However, once surgery is completed, one needs to restart the Ricci flow process, at which point further singularities can occur. In order to apply surgery to these further singularities, we need to check that all the properties we have been exploiting about Ricci flows – notably the Hamilton-Ivey pinching property, the -noncollapsing property, and the existence of canonical neighbourhoods for every point of high curvature – persist even in the presence of a large number of surgeries (indeed, with the way the constants are structured, all quantitative bounds on a fixed time interval [0,T] have to be uniform in the number of surgery times, although we will of course need the set of such times to be discrete). To ensure that surgeries do not disrupt any of these properties, it turns out that one has to perform these surgeries deep in certain -horns of the Ricci flow at the singular time, in which the geometry is extremely close to being cylindrical (in particular, it should be a -neck and not just a -neck, where the surgery control parameter is much smaller than ; selection of this parameter can get a little tricky if one wants to evolve Ricci flow with surgery indefinitely, although for the purposes of the Poincaré conjecture the situation is simpler as there is a fixed upper bound on the time for which one needs to evolve the flow). Furthermore, the geometry of the manifolds one glues in to replace the disappearing regions has to be carefully chosen (in particular, it has to not disrupt the pinching condition, and the geometry of these glued in regions has to resemble a -cap for a significant amount of (rescaled) time). The construction of the “standard solution” needed to achieve all these properties is somewhat delicate, although we will not discuss this issue much here.

In this, the final lecture, we shall present these issues from a high-level perspective; due to lack of time and space we will not cover the finer details of the surgery procedure. More detailed versions of the material here can be found in Perelman’s second paper, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu. (See also a forthcoming paper of Bessières, Besson, Boileau, Maillot, and Porti.)

– Ricci flow at the singular time –

Suppose we have a compact 3-dimensional Ricci flow on the time interval without any embedded with trivial normal bundle; for simplicity we can take M to be connected (otherwise we simply treat each of the finite number of connected components of M separately). We are interested in the extent to which we can define a limiting geometry g(T) on M (or on some subset of M) at the final time T, and to work out the topological structure of the portions of M for which such a limit cannot be defined.

From Theorem 1 of Lecture 18, we know that any point for which the curvature R(t,x) exceeds a certain threshold K, will lie in a canonical neighbourhood. (For sake of discussion we shall suppress the constants C and , as they will not play a major role in what follows.) One consequence of this is that one has the pointwise bounds

(1)

whenever . Also recall from the maximum principle that we have throughout.

These simple regularity properties of the scalar curvature R are already enough to classify the limiting behaviour of as for each fixed x:

Exercise 1. Using (1), show that for every there are either two possibilities: either remains bounded as (with a bound that can depend on x), or that goes to infinity as , and in the latter case we even have the stronger statement for some c depending only on the implied constant in (1). If we let be the set of x for which remains bounded, show that is open, and converges uniformly on compact subsets of to some limit as .

The pinching property also lets us establish bounds of the form . Using this and Shi’s regularity estimates (and the non-collapsing property), one can show that converges in on compact subsets of to an incomplete limit .

Our main tasks here are to understand the geometry of the limit , and the topology of the remaining region (and how the two regions connect to each other).

If is all of M, then the Ricci flow continues smoothly to time T, and we can continue onwards beyond T by the local existence theory for that flow. Now let us instead consider the other extreme case in which is empty. In this case, from Exercise 1 we see that we have for all , if t is sufficiently close to T. In particular, this means that every point in M lies in a canonical neighbourhood: an -round component (topologically ), a C-component (topologically or ), a -neck (topologically ), or a -cap (topologically a 3-ball or punctured ). If any point lies in the first type of canonical neighbourhoods, then M is topologically a spherical space form . Similarly, if any point lies in the second type, M is either an or this way. So the only remaining case left is when every point lies in a neck or a cap. Since each cap contains at least one neck in it, we have at least one neck; following this neck in both directions, we either must end up with a doubly capped tube, or the tube must eventually connect back to itself. In the former case we obtain an , , or (depending on whether zero, one, or two of the caps are punctured ‘s rather than 3-balls); in the latter case, we get an bundle over , which as discussed back in Lecture 2 comes in only two topological types, oriented and unoriented.

To summarise, if is empty, then M is either a spherical space form,
, or an bundle over . In this case, the surgery procedure is simply to delete the entire manifold; this respects the topological compatibility condition required for Theorem 2 of Lecture 2. (The geometric compatibility condition is moot in this case.) In this case, the disappearing region is the whole manifold M, and the continuing region is empty.

Similar considerations occur if is non-empty, but that (say) for all . So we may assume that there is at least one x for which , and thus for all t sufficiently close to T. Thus we are guaranteed at least one point of bounded curvature in M, even at times close to the singular time. We can also assume that no canonical neighbourhood in M is an -round or C-component, since again in this case we could delete the entire manifold by surgery. Thus every point of curvature greater than K lies in a neck or a cap.

Because of this, it is not hard to show that every boundary point of (where becomes infinite) lies at the end of an -horn:a tube of -necks of curvature at least 4K (say) throughout, with the curvature becoming infinite at one or both ends (thus the width of the necks go to zero as one approaches the boundary of ). (Note that if the tube is ever capped off by a -cap, then the curvature does not go to infinity in this tube.) If the curvature goes to infinity at both ends, we have a double -horn; otherwise, we have a single -horn will have one infinite curvature end and one end with bounded curvature.

The single -horns are all disjoint from each other, and their volume is bounded from below, and so they are finite in number. So the geometric picture of is that of a (possibly infinite) number of double -horns, together with a finite number of additional connected incomplete manifolds, with boundary consisting of a finite number of disjoint spheres , with a single -horn glued on to each one of these spheres.

Suppose one performs a topological surgery on each single -horn, by taking a sphere somewhere in the middle of each horn, removing the portion between that and the boundary, and replacing it by a 3-ball. We also remove all the double -horns; all the removed regions form the disappearing region of M, and the remainder is the continuing region. This creates a new compact (but possibly disconnected) manifold , formed by gluing finitely many 3-balls to the continuing region. To see the topological relationship between this new manifold and the previous manifold M, we move backwards in time a slight amount to an earlier time t, so that the horn is no longer singular at its boundary and instead connects to the remainder of the manifold. If t is close enough to T, then (by (1)), the portion of the horn between the and the boundary of the horn will still have curvature at least K, and thus every point here will lie in a neck or cap. Also, all the points in , and in particular the portion of the manifold beyond the boundary of the horn, will also have curvature at least K and thus lie in a neck or cap if t is close enough to T. If we then follow this desingularised horn from the surgery sphere towards its boundary and beyond (possibly passing through any number of double -horns in the process), we will either discover a capped tube (which is thus topologically either a 3-ball or a punctured ), or else the tube will eventually connect to another surgery sphere, which may or may not lie in the same connected component of M(T). Topologically, the first case corresponds to taking a connected sum of (one component of) M(T) with either a sphere or a projective space ; the second case corresponds to taking a connected sum of one component of M(T) with either another component of M(T), or with an -bundle over . Putting all this together we see that M is the connected sum of the components of M(T), together with finitely many ‘s, ‘s, and -bundles over , which again gives the topological compatibility condition required for Theorem 2 of Lecture 2.

We have thus successfully performed a single (topological) surgery. However, in doing so we have lost a lot of quantitative properties of the geometry, such as Hamilton-Ivey pinching, -noncollapsing, and the canonical neighbourhood property, which means that we cannot yet ensure that we can perform any further surgeries. To resolve this problem, we need to be more precise about the surgery process, in particular using our freedom to choose the surgery sphere as deep inside the -horn as we please, and to prescribe the metric on the cap that we attach to that sphere.

– Surgery –

To do surgery, a key observation of Perelman is that the geometry of the horn becomes increasingly cylindrical as one goes deeper into the horn:

Lemma 1. Let H be a single -horn which has width scale comparable to r at the finite curvature end, and let . Then there exists a -neck of width scale comparable to h inside the horn H, where is a small quantity depending only on r and .

Proof. (Sketch) Suppose this was not the case; then one could find a sequence of horns of this type, and a sequence of points inside these horns inside -necks of width scale comparable to which are not inside -necks of this scale, where . We can find a minimising geodesic from the finite curvature end to the infinite curvature end that goes through the neck near . We then rescale to have width 1 at , and then apply the machinery from Lecture 18 to obtain a limit ; the bounded curvature at bounded distance property (Proposition 3 from Lecture 18) shows that both the bounded curvature end and the infinite curvature end of the horn must recede to be infinitely far away from in the limit, and so becomes complete; it also has non-negative curvature, by pinching. The minimising geodesic becomes a minimising line in , and so by the Cheeger-Gromoll theorem it splits into the product of a line and a two-dimensional manifold (which is -close to a sphere). It turns out that we can continue all these manifolds backwards in time and repeat these arguments (much as in Lecture 18) to eventually give the structure of a -solution; but then the vanishing curvature forces it to be a round cylinder, by Proposition 2 of Lecture 17. This implies that the rescaled are eventually -necks, a contradiction.

In order to successfully perform Ricci flow with surgery up to some specified time T (starting from controlled initial conditions, and as always assuming that no embedded with trivial normal bundle is present), we shall pick to be a very small number depending on T and the initial condition parameters, and perform our surgery on the -necks the scale h provided by Lemma 1, where is (essentially) the curvature threshold beyond which the canonical neighbourhood condition holds. (In order to avoid a circular dependence of constants, one needs to check that even after surgery, that the curvature threshold for the canonical neighbourhood condition remains bounded even after arbitrarily many surgeries, as long as is chosen sufficiently small depending on this scale, on T, and on the initial conditions.)

Remark 1. Thanks to finite time extinction in the simply connected case, being able to perform Ricci flow with surgery up to a preassigned finite time T is sufficient for proving the Poincaré conjecture (cf. Remark 1.4 of Perelman’s third paper). For the full geometrisation conjecture, however, one needs to perform Ricci flow with surgery for an infinite amount of time. For this, one cannot pick a single ; instead, one has to divide the time interval into bounded intervals (e.g. dyadic intervals), and pick a different for each one (which depends on a number of parameters, including the curvature threshold for the canonical neighbourhbood property on the previous dyadic interval). This selection of constants becomes a little subtle; see e.g. the notes of Kleiner-Lott for further discussion.

Having located a -neck inside each single -horn, we remove the half of the neck from the centre sphere to the infinite curvature region, and smoothly interpolate in its place a copy of an (appropriately rescaled) standard solution. There is some choice as to how to set up this solution (much as there is some freedom when selecting a cutoff function), but roughly speaking this solution should resemble the manifold formed by attaching a hemispherical cap to a round unit cylinder, except that one needs to smooth out the transition between the two portions of this solution; also, one needs to ensure that one has positive curvature throughout the standard solution in order not to disrupt the Hamilton-Ivey pinching property. It is also technically convenient to demand that this solution is spherically symmetric (at which point Ricci flow collapses to a system of two scalar equations in one spatial dimension). One can show that such standard solutions exist for unit time (just as the round unit cylinder does), and asymptotically matches the round shrinking solution at spatial infinity. As one consequence of this, one can check that all points in spacetime on the standard solution have canonical neighbourhoods; and, with some effort, one can also show that the same will be true in the spacetime vicinity of the region in which a standard solution has been inserted via surgery into a Ricci flow, as long as is sufficiently small. This is an essential tool to ensure that the canonical neighbourhood solution persists after multiple surgeries.

Remark 2. Suppose that M is irreducible with respect to connected sum (one can easily reduce to this case for the purposes of the Poincaré conjecture, thanks to Kneser’s theorem on the existence of the prime decomposition). Then all surgeries must be topologically trivial, which means that every -horn, when viewed just before the singular time, only connects to a tube capped off with a ball. Then one can show that the surgery procedure is almost distance decreasing in the sense that for any , there exists a -Lipschitz diffeomorphism from the pre-surgery manifold to the post-surgery manifold. This property is useful for ensuring that various arguments for establishing finite time extinction for Ricci flows, also work for Ricci flows with surgery, as discussed for in Lecture 5 and Lecture 6. Even if the manifold is not irreducible, one can show that there are only finitely many surgeries that change the topology of the manifold; this can be established either using the prime decomposition, or by constructing a topological invariant (namely, the maximal number of homotopically non-trivial and homotopiclly distinct embedded 2-spheres in M) which is finite, non-negative, decreases by at least one with non-trivial surgery; see Section 18.2 of Morgan-Tian for details.

Remark 3. The various properties listed above of the standard solution and its insertion into surgery regions are the “geometric compatibility conditions” alluded to in Theorem 2 of Lecture 2.

– Controlling the geometry after multiple surgeries –

Suppose that we have already performed a large (but finite) number of surgeries. In order to be able to continue Ricci flow with surgery, it is necessary that we maintain quantitative control on the geometry of the manifold which is uniform in the number of surgeries. Specifically, we need to extend the following existing controls on Ricci flow, to Ricci flow with surgery:

1. Lower bounds on .
2. Hamilton-Ivey pinching type bounds that lower bound in terms of R.
3. -noncollapsing of the manifold.
4. Canonical neighbourhoods for all high curvature points in the flow.

The first two controls are quite easy to establish, because they are propagated by Ricci flow (thanks to the maximum principle), and are easily preserved by surgery (basically because 1. and 2. are primarily concerned with negative curvature, and surgery is only performed in regions of high positive curvature by construction). It is significantly trickier however to preserve 3., because the proof of -noncollapsing is more global, requiring the use of -geodesics through spacetime. The key new difficulty is that thanks to the presence of surgery, the manifold can become “parabolically disconnected”; not every point in the initial manifold can be reached from a future point in a later manifold by an -geodesic, because an intervening surgery could have removed the region of spacetime that the geodesic ought to have passed through. This forces one to introduce the notion of an admissible curve – curves that avoid the surgery regions completely – and barely admissible curves, which are admissible curves which touch the boundary of the surgery regions. Roughly speaking, the monotonicity of reduced volume now controls the -noncollapsing at future times in terms of the non-collapsing of the portion of the initial manifold which can be reached by admissible curves; this region is bordered by points which can be reached by barely admissible curves.

Now suppose we knew that all barely admissible curves had large reduced length. Then the maximum principle argument that located points of small reduced length for Ricci flows (cf. equation (18) from Lecture 11), would continue to work for Ricci flows with surgery, with the points located being inside the admissible region. It turns out that the arguments of Lecture 11 could then be adapted to this setting without much difficulty to establish the desired -noncollapsing.

It is not too difficult to show that if a path did pass through a -neck inside an -horn in which surgery was taking place, then the portion of the path near to that surgery region would have a large contribution to the reduced length (unless the starting point of the path was very close to the surgery region, but then one could verify the non-collapsing property directly, essentially due to the non-collapsed nature of the standard solution). This almost settles the problem immediately, except for the technical issue that there might be regions of negative curvature elsewhere in spacetime which could drag the reduced length back down again (note that the reduced length is not guaranteed to be non-negative!). There is a technical fix for this, defining a modified reduced length in which the curvature term R is replaced by (and using the lower bounds on to measure the discrepancy between the two notions), but we will not discuss the details here; see Lemma 5.2 of Perelman’s second paper (and Chapter 16 of Morgan-Tian for a very detailed treatment).

Remark 4. A recent paper of Zhang uses Perelman’s entropy (as in Lecture 8) to establish -noncollapsing for Ricci flow with surgery, using the distance-decreasing property to keep control of the entropy functional after each (topologically trivial) surgery. This should provide a way to simplify this part of the argument, at least in the case of irreducible manifolds M.

Finally, one has to check that all high-curvature points of Ricci flows with surgery lie in canonical neighbourhoods, where the threshold for “high curvature” is uniform in the number of surgeries performed. Very roughly speaking, there are two cases, depending on whether there was a surgery performed near (in the spacetime sense) such a region or not. If there was no nearby surgery, then the arguments in Lecture 18 (which are local in nature) essentially go through, exploiting heavily the -noncollapsing and pinching properties that we have just established. If instead there was a nearby surgery in the recent past, then one needs to approximate the geometry here by the geometry of the standard solution, for which all points have canonical neighbourhoods. See for instance Section 17.1 of Morgan-Tian for details.

– Surgery times do not accumulate –

The very last thing one needs to do to establish the Poincaré conjecture is to establish Theorem 3 from Lecture 2, which asserts that the set of surgery times is discrete. It turns out that this is in fact rather easy to establish. One first observes that each surgery removes at least some constant amount of of
volume from the manifold (as can be seen by looking at what happens to a single -neck of width roughly h under surgery; all other removals under surgery of course only decrease the volume further). On the other hand, using the volume variation formula (equation (33) from Lecture 1) we have an upper bound on the growth of volume during non-surgery times:

. (2)

Since we have a uniform lower bound on , this implies that volume can grow at most exponentially, and in particular can only grow by a bounded amount on any fixed time interval. Hence there can be at most finitely many surgeries on each such time interval, and we are done.

Remark 5. The number of surgeries performed in a given time interval, while finite, could be incredibly large; it depends on the length scale h of the surgery, which in turn depends on the parameter , which needs to be very small in order not to disrupt the -noncollapsing or canonical neighbourhood properties of the flow. This is why it is essential that our control of such properties is uniform with respect to the number of surgeries.

Remark 6. Note also that there is no lower bound as to how close two surgery times could be to each other; indeed, there is nothing preventing two completely unrelated surgeries from being instantaneous. However, if there are an infinite number of singularities occurring at (or very close to) a single time, what tends to happen is that the earliest surgeries will not only remove the immediate singularities being formed, but will also pre-emptively eradicate a large number of potential future singularities (in particular, due to the removal of all the double -horns, which were not immediately singular but were threatening to become singular very shortly), thus keeping the surgery times discrete.

This concludes the lecture notes on the Poincaré conjecture. Have a good summer!

The material here is drawn largely from Morgan-Tian’s book and Perelman’s first and second papers; see also Kleiner-Lott’s notes and Cao-Zhu’s paper for closely related material. Due to lack of time, some details here may be a little sketchy.

– Canonical neighbourhoods –

Let us formally define the notions of a canonical neighbourhood that were introduced in Lecture 12. They are associated with the various local geometries that are possible for three-dimensional -solutions. The first type of neighbourhood is related to the round spherical space forms .

Definition 1. (-round) Let . A compact connected 3-manifold is -round if one can identify M with a spherical space form with the constant curvature metric h such that some multiple of g lies within of h in the topology.

Note that if a sequence of manifolds , after rescaling, is converging geometrically to a spherical space form, then for any such manifolds will be -round for sufficiently large n.

The next type of canonical neighbourhood is associated to the small compact manifolds from Proposition 9 of the previous lecture.

Definition 2. (C-component) Let . A C-component is a connected 3-manifold homeomorphic to or , such that after rescaling the metric by a constant, the sectional curvatures, diameter, and volume are bounded between 1/C and C, and first and second derivatives of the curvature are also bounded by C.

[We have deviated slightly here from the definition in Morgan-Tian by adding control of first and second derivatives for minor technical reasons.]

Thus, for instance, every compact -solution which is small but not round in the sense of Proposition 9 will be a C-component for some C. Also observe that if a sequence of manifolds converge geometrically to a C-component, then these manifolds will be (say) 2C-components once one is sufficiently far along the sequence.

The remaining canonical neighbourhoods are incomplete, corresponding to portions of non-compact (or compact but large) -solutions. One of them is the -neck defined in Definition 1 of Lecture 17. The other is that of a cap.

Definition 3. (-cap) Let . A -cap is an incomplete 3-manifold that is the union of an -neck N with an incomplete core along one of the boundaries of the neck N. The core is homeomorphic to or a punctured , and has boundary equal to the above boundary of N. Furthermore, after rescaling g by a constant, the sectional curvatures, diameter, volume in the core are bounded between 1/C and C, and the zeroth, first and second derivatives of curvature in the cap are bounded above by C.

Definition 4. (Canonical neighbourhood) Let . We say that a point x in a 3-manifold (M,g) (possibly disconnected) has a -canonical neighbourhood if one of the following is true:

1. x lies in an -round component of M.
2. x lies in a C-component of M.
3. x is the centre of an -neck in M.
4. x lies in the core of a -cap in M.

We remark that if a sequence of pointed manifolds converges to a limit , and has a -neighbourhood of , then for sufficiently large n, has a -neighbourhood (say) of . Also observe from construction that the property of having a canonical neighbourhood is scale-invariant.

Exercise 1. (First derivatives of curvature) Show that if is sufficiently small, and x has a -canonical neighbourhood, then R(x) is positive, , and .

From the theory of the previous lecture we have

Proposition 1. For every there exists such that every point in a 3-dimensional -solution at any given time will have a -canonical neighbourhood, unless it is a round shrinking .

Note that C and are independent of ; this is thanks to the universality property (Proposition 5 from Lecture 17).

Remark 1. For technical reasons, one actually needs a slightly stronger version of this proposition, in which any canonical neighbourhood which is an -neck is extended backwards to some extent in time in a manner that preserves the neck structure (leading to the notion of a strong -neck and strong canonical neighbourhood); see Chapter 9.8 of Morgan-Tian for details. Technically, the results below need to be stated for strong canonical neighbourhoods

The objective of this lecture is to establish the analogous claim for high-curvature regions of arbitrary Ricci flows:

Theorem 1. (Structure of high-curvature regions) For every there exists such that the following holds. Let be a three-dimensional compact Ricci flow on some time interval [0,T) with no embedded with trivial normal bundle. Then there exists such for every time , every with has a -canonical neighbourhood.

This theorem will then allow us to perform surgery on Ricci flows, as we will discuss in the final lecture.

Morally speaking, Theorem 1 follows from Proposition 1 by rescaling and compactness arguments, but there is a rather delicate issue involved, namely to gain enough control on curvature at points in spacetime both near (and far) from the chosen point (t,x) that the Hamilton compactness theorem can be applied.

-- Overview of proof --

We begin with some reductions. We can of course take M to be connected. Fix , and take C sufficiently large depending on (but not depending on any other parameters). We first observe that it suffices to prove the theorem for closed intervals [0,T] rather than half-open ones, as long as the bounds on K depend only on an upper bound on T and the initial metric g(0) and not on T itself (in particular, one cannot just use the trivial fact that R will be bounded on any compact subset of spacetime such as .) Once one does this, one sees that Theorem 1 is now true for some enormous K that depends on T; the task is to get a uniform K that depends only on the initial metric g(0) and on an upper bound for T.

Perelman’s argument proceeds by a downward induction on K; assume that K is large (depending on g(0) and ), and that Theorem 1 has already been established for 4K (say); and then establish the claim for K. By the previous discussion, this conditional result will in fact imply the full theorem.

By rescaling we may assume that g(0) has normalised initial conditions (curvature bounded in magnitude by 1, volume of unit balls bounded below by some positive constant ). We will now show that the conditional version of Theorem 1 holds for K sufficiently large depending only on and .

Suppose this were not the case. Then there would be a and a , and a sequence of pointed Ricci flows on (not containing any embedded with trivial normal bundle) for some with normalised initial conditions with constant , times , and scalars such that every point in of scalar curvature at least 4K_n has a -canonical neighbourhood, but that but does not have a -canonical neighbourhood (in particular, ). We want to extract a contradiction from this.

From the local theory (Lemma 1 from Lecture 11) we know that the curvature is bounded for short times (t less than a universal constant depending only on ), so must be bounded uniformly from below.

As usual, we define the rescaled pointed flows by , , and . Thus these flows are increasingly ancient and have scalar curvature between 1 and 4 at the origin . Also, any point in these flows of curvature at least 4 is contained in a canonical neighbourhood.

By Perelman’s non-collapsing theorem (Theorem 2 from Lecture 7), we know that the flows flow is -noncollapsed at all scales less than 1 (say) for some depending only on ; by rescaling, the rescaled flows are then -noncollapsed at all scales less than 1/o(1).

Meanwhile, from the Hamilton-Ivey pinching theorem (Theorem 1 from Lecture 3) we have and whenever . Rescaling this, we obtain and .

Suppose we were able to prove the following statement.

Proposition 2. (Asymptotically globally bounded normalised curvature) For any we have a bound for all and , if n is sufficiently large depending on .

From this and the -noncollapsing, we see that the Hamilton compactness theorem (Theorem 2 from Lecture 15) applies, and after passing to a subsequence we see that the pointed flows converges geometrically to a Ricci flow which has bounded scalar curvature on for each , and is automatically connected, complete, and ancient, and without an embedded with trivial normal bundle. From pinching we also see that we have non-negative sectional curvature; from the -noncollapsing of the flows we have -noncollapsing of the limiting flow . From Hamilton’s Harnack inequality (cf. Lecture 13) we can show , and so we in fact have globally bounded curvature. Finally, since is bounded between 1 and 4, so is ; thus the flow is not flat. Putting all this together, we conclude that is a -solution (see Definition 1 from Lecture 12). From Proposition 1, has a -canonical neighbourhood in (if C is chosen large enough depending on ); thus will have a -canonical neighbourhood in for large enough n, and so by rescaling has a -canonical neighbourhood in , contradicting the hypothesis, and we are done.

So it remains to prove Proposition 2. If we had the luxury of picking to be a point which had maximal curvature amongst all other points in , then this proposition would be automatic. However, we do not have this luxury (roughly speaking, this would only let us get canonical neighbourhoods for the “highest curvature region” of the Ricci flow, leaving aside the “second highest curvature region”, “third highest curvature region”, etc., unprepared for surgery). So one has to work significantly harder to achieve this aim.

– Bounded curvature at bounded distance –

A key step in the execution of Proposition 2 is the following partial result, in which the bound on curvature is allowed to depend on A, and for which one cannot go backwards in time.

Proposition 3. (Bounded curvature at bounded distance) For any A > 0 we have a bound for all , if n is large enough depending on A.

This partial result is already rather tricky; we sketch the proof as follows (full details can be found in Chapter 10 of Morgan-Tian, Section 51 of Kleiner-Lott, or Section 7.1 of Cao-Zhu). If this result failed, then we have a sequence with bounded and , thus one can move a bounded distance along a minimising geodesic from (which has curvature between 1 and 4) to and reach a point of arbitrarily high curvature. On the other hand, we know that every point of curvature at least 4 has a canonical neighbourhood. Thus there is a bounded length minimising geodesic in that goes entirely through canonical neighbourhoods, starts with scalar curvature 4, and ends up with arbitrarily high curvature, with curvature staying 4 or greater throughout this process. This cannot happen if the canonical neighbourhoods are -round or C-components (since these neighhourhoods are already complete and curvatures are comparable to each other on the entire neighbourhood), so this geodesic can only go through -necks and -caps. One can also rule out the latter possibility (a long geodesic path that goes through the core of a -cap can easily be shown to not be minimising); thus the geodesic is simply going through a tube of -necks, with the width of these necks starting off being comparable to 1 and ending up being arbitrarily small. It turns out that by using a version of Hamilton’s compactness theorem for incomplete Ricci flows, one can take a limit, which at time zero is a tube (topologically ) of non-negative curvature in which the curvature has become infinite at one end. Also, thanks to time derivative control on the curvature (see Exercise 1), the tube can be extended a little bit backwards in time as an incomplete Ricci flow (though the amount to which one can do this shrinks to zero as one approaches the infinite curvature end of the tube).

One can show that as one approaches the infinite curvature end of the cylinder and rescales, the cylinder increasingly resembles a cone. (For instance, one can use the bound from Lemma 1 of Lecture 15, where are geodesics emanating from the infinite curvature end, to establish this sort of thing.) By taking another limit one can then get an incomplete Ricci flow which at time zero is a cone. Because curvature is bounded away from zero, this cone is not flat. At this point, a version of Hamilton’s splitting theorem (Proposition 1 from Lecture 13) for incomplete flows asserts that the manifold locally splits as the product of a line and a two-dimensional manifold. But non-flat cones cannot split like this, a contradiction. This establishes Proposition 3.

Remark 2. More generally, this argument can be used to show that if is bounded by some , then is bounded by for all .

– Bounded curvature at all distances –

Now we extend Proposition 3 by making the bound global in A:

Proposition 4. (Bounded curvature at all distances) For any A > 0 we have a bound for all , if n is large enough depending on A.

We sketch a proof as follows. From Proposition 3 and compactness (taking advantage of non-collapsing, of course) we already know (passing to a subsequence if necessary) that converges to some limit
which has non-negative curvature; it can also be extended a little bit backwards in time as an incomplete Ricci flow. Also, every point in this limit of curvature greater than 4 has a canonical neighbourhood. Our task is to basically to show that has bounded curvature. If this is not the case, then there are points of arbitrarily high curvature, which must be contained in either -necks or -caps. We conclude that there exist arbitrarily narrow -necks. One can then show that the manifold had strictly positive curvature, since otherwise by Hamilton’s splitting theorem the manifold would split locally into a product of a two-dimensional manifold and a line, which can be shown to be incompatible with having arbitrarily narrow necks.

At this point one uses a general result that complete manifolds of strictly positive curvature cannot have arbitrarily narrow necks. We sketch the proof as follows. Clearly we may assume the manifold is compact, and hence by the soul theorem is diffeomorphic to . This implies that every neck in fact separates the manifold into a compact part and a non-compact part. In fact, one can show that if p is a soul for the manifold, then there is a minimising geodesic from p to infinity that passes through all the necks. But if one then considers the Busemann function , one can show that the gradient field is a unit vector which is within to parallel to the necks. This, combined with Stokes theorem, tells us that the area of the level sets of B inside a neck (which, up to errors of , are basically slices of that neck) does not fluctuate by more than , even as one compares very distant necks together. But this contradicts the assumption that there are arbitrarily small necks. (For full details see Proposition 2.19 of Morgan-Tian. )

– Bounded curvature at all times –

Now we need to extend Proposition 4 backwards in time. The time derivative bound on curvature (Exercise 1) lets us extend backwards by some fixed amount of time, but at the cost of potentially increasing the curvature, and we cannot simply iterate this (much as one cannot iterate a local existence result for a PDE to obtain a global one without some sort of a priori bound on whatever is controlling the time of existence). But what Exercise 1 does let us do, is reduce matters to establishing an a priori bound:

Proposition 5. (A priori bound) Let , and suppose is uniformly bounded on for all sufficiently large n. Then in fact we can bound on these slabs by a universal bound (not depending on the previous universal bound).

Indeed, Exercise 1 then lets us extend the uniform bounds a little bit to the past of , and one can continue this procedure indefinitely to establish Proposition 2.

We sketch the proof as follows. We allow all implied constants to depend on for brevity. The bounds are already enough to give a non-ancient limiting flow on which is complete, connected, and non-negative curvature which is bounded at all times (but with an unspecified bound), and bounded at time zero by . Also, every point with curvature greater than 4 is known to have a canonical neighbourhood. The challenge is now to propagate the quantitative curvature bounds backwards in time, to replace the qualitative bound.

In the case of an ancient flow of non-negative curvature, Hamilton’s Harnack inequality (equation (29) from Lecture 13) gives , which automatically does this propagation for us. We are however non-ancient here, and the Harnack inequality in this setting only gives a bound of the form . This can be integrated to give , thus our bounds blow up as we approach . However, this is at least enough to get good control on distances; in particular, using Corollary 1 from Lecture 15 we see that

(1)

for all . Fortunately, the left-hand side here is absolutely integrable, and so we obtain a useful global distance comparison estimate:

. (2)

To use this, pick a large curvature , then a much larger radius r, then an extremely large curvature L’. Now suppose for contradiction that we have a point (t,x) in of curvature larger than L’. This point is then contained in a canonical neighbourhood. This neighbourhood cannot be compact (i.e. an -round or C-component), since that would mean that the
minimal scalar curvature was comparable to L’ at time t, which by monotonicity of (Proposition 2 of Lecture 3) would mean that the scalar curvature is comparable to L’ at time 0, contradicting the boundedness of curvature there. This argument in fact shows that all large curvature regions are contained in either -necks or -caps.

Consider the ball . From Remark 2 we see (if L’ is large enough) that the curvature is larger than L on this ball, and so this ball consists entirely of necks and caps of width at most . From this it is not hard to see that the volume of this ball at time t is . On the other hand, there must be at least one point y on the boundary of this ball, since otherwise would be at least L, which as noted before is not possible.

Applying (2) (and noting that Ricci flow reduces volume when there is non-negative curvature, see equation (33) of Lecture 2) we conclude that also has volume O(L^{-1/2} r). On the other hand, we know that there is a point y at distance r from x at time t, thus y at distance r-O(1) from x at time 0. Thus (by the triangle inequality, and dividing the geodesic from x to y at time zero into unit length segments) contains disjoint balls of radius 1/2 (say). By the non-collapsing and curvature bounds at time zero, this forces to have volume at least comparable to r, a contradiction. This proves Proposition 5 and thus Theorem 1.

Remark 3. Perelman (and the authors who follow him) uses a slight variant of this argument, using the soul theorem to fashion a small in a narrow neck that separates two widely distant points at time t, which then evolves to a small separating two widely distant points at time zero (here we use (2)). But this is not possible due to the bounded curvature at that time.

The arguments are particularly simple when the asymptotic gradient shrinking soliton is compact; in this case, the rounding theorems of Hamilton show that the -solution is a (time-shifted) round shrinking spherical space form. This already classifies -solutions completely in two dimensions; the only remaining case is the three-dimensional case when the asymptotic gradient soliton is a round shrinking cylinder (or a quotient thereof by an involution). To proceed further, one has to show that the -solution exhibits significant amounts of curvature, and in particular that one does not have bounded normalised curvature at infinity. This curvature (combined with comparison geometry tools such as the Bishop-Gromov inequality) will cause asymptotic volume collapse of the -solution at infinity. These facts lead to the fundamental Perelman compactness theorem for -solutions, which then provides enough geometric control on such solutions that one can establish the structural theorems mentioned earlier.

The treatment here is a (slightly simplified) version of the arguments in Morgan-Tian’s book, which is based in turn on Perelman’s paper and the notes of Kleiner-Lott (see also the paper of Cao-Zhu for a slightly different treatment of this theory).

– The compact soliton case –

As we saw in Lecture 15, every -solution has at least one asymptotic gradient shrinking soliton associated to it. Suppose we are in the case in which at least one of these asymptotic gradient shrinking solitons is compact; by Theorem 1 of Lecture 16, this means that this soliton is a round shrinking spherical space form. Since this soliton is the geometric limit of a rescaled sequence of M, this implies that M is homeomorphic to and, along a sequence of times , converges geometrically after rescaling to a round spherical space form. Thus M is asymptotically round as .

One can now apply Hamilton’s rounding theorems in two and three dimensions to conclude that M is in fact perfectly round. In the case of two dimensions this can be done by a variety of methods; let me sketch one way, using Perelman entropy; this is not the most elementary way to proceed but allows us to quickly utilise a lot of the theory we have built up. First we can lift M up to be instead of the quotient . Then we observe from the Gauss-Bonnet theorem (Proposition 1 from Lecture 4) that , and hence by the volume variation formula (equation (33) from Lecture 1) the volume is decreasing in time at a constant rate . Let us shift time so that the volume is in fact equal to , and consider the Perelman entropy defined in Lecture 8. Testing this entropy with ) we obtain an upper bound . On the other hand, on the sequence of times , is smoothly approaching a round sphere, on which the entropy can be shown to be exactly by the log-Sobolev inequality for the sphere (which can be proven in a similar way to the log-Sobolev inequality for Euclidean space in Lecture 8). Thus one can soon show that . On the other hand, this entropy is non-increasing in ; thus is constant. Applying the results from Lecture 14 we conclude that this time-shifted manifold M is itself a gradient shrinking soliton, and thus is round by the results of Lecture 15.

Exercise 1. In this exercise we give an alternate way to establish the roundness of M in two dimensions, using a slightly different notion of “entropy”. Firstly, observe that under conformal change of metric on a surface, one has , , and . If we then express where h is the metric on of constant curvature +1, show that the Ricci flow equation becomes , and in particular that the volume is decreasing at constant rate . If we time shift so that , show that the relative entropy is non-decreasing in , and converges to 0 along (here one needs a stability result for the uniformisation theorem). From this and the converse to Jensen’s inequality, conclude that a is constant at every time, which gives the rounding. (For more proofs of the rounding theorem, for instance using the Hamilton entropy , see the book of Chow and Knopf.)

In two dimensions, we saw in the previous lecture that the only gradient shrinking soliton was the round shrinking sphere. We have thus shown the following classification of -solutions in two dimensions:

Proposition 1. The only two-dimensional -solutions are time translates of the round shrinking and .

For three dimensions, we can argue as in Case 4 of the previous lecture. Write for the eigenvalues of the curvature tensor. At the times , we have for some . Applying the tensor maximum principle (Proposition 1 from Lecture 3) and the analysis from Case 4 of the previous lecture, we thus see that for all times ; sending n to infinity we conclude that for all times, and so curvature is conformal. Using the Bianchi identity as in Case 4 of the previous lecture, we conclude that the manifold is round.

– The case of a vanishing curvature –

Now we deal with the case in which there is a vanishing curvature:

Proposition 2. Let be a 3-dimensional -solution for which the Ricci curvature has a null eigenvector at some point in spacetime. Then M is a time-shifted round shrinking cylinder, or the oriented or unoriented quotient of that cylinder by an involution.

Proof. If the Ricci curvature vanishes at any point, then by Hamilton’s splitting theorem (Proposition 1 from Lecture 13) the flow splits (locally, at least) as a line and a two-dimensional flow. Passing to a double cover if necessary, we see that the flow is the product of a two-dimensional Ricci flow and either a line or a circle. The two-dimensional flow is itself a -solution and is thus a round shrinking or . Checking all the cases and eliminating those which are not -noncollapsed we obtain the claim.

– Asymptotic volume collapse –

Our next structural result on -solutions is

Proposition 3. (Asymptotic collapse of Bishop-Gromov reduced volume) Let be a -solution of dimension 3. Then for any time t and , .

Proof. We first observe, by inspecting all the possibilities from Theorem 1 of Lecture 16, that the claim is already true of all 3-dimensional asymptotic gradient shrinking solitons. We apply this to a gradient shrinking soliton for M and conclude that for any there exists arbitrarily negative times , points and radii such that . Applying the Bishop-Gromov comparison inequality (Lemma 1 from Lecture 9) we conclude that . By the triangle inequality this implies that .

Now we need to move from time to time t; since is arbitrarily negative we can assume . Recall from Lemma 1 of Lecture 15 and the bounded curvature hypothesis that is bounded for all times and all geodesics . Plugging this into the Ricci flow equation, we see that is also bounded (in the sense of forward difference quotients) for all times and all geodesics. In particular we have the additive distance fluctuation estimate , where the error is bounded even as or goes to infinity. Also, from equation (33) from Lecture 1 we know that the volume measure is decreasing over time. From this we conclude that . Since is arbitrary, the claim follows.

We have a corollary:

Corollary 1. Let be a non-compact -solution of dimension 3. Then for any time t and point we have . (Of course, the claim is vacuous for compact solutions.)

Proof. By time shifting we may take t=0. Suppose for contradiction that is finite, thus at time t=0, and thus at all previous times since (equation (29) of Lecture 13). From the non-negativity of the curvature we obtain the similar upper bounds on the Riemann curvature. From the -noncollapsed nature of M we may thus conclude that is bounded away from zero for some small c > 0. But this contradicts Proposition 3.

Remark 1. In other treatments of this argument (e.g. in Morgan-Tian), Corollary 1 is established first (using the Topogonov theory from Lecture 16) and then used to derive Proposition 3. The two approaches are essentially just permutations of each other, but the arguments above seem to be slightly simpler (in particular, the theory of the Tits cone is avoided).

By combining Proposition 3 with another compactness argument, we obtain an important relationship:

Corollary 2. (Volume noncollapsing implies curvature bound) Let be a 3-dimensional -solution, and let be a ball at time zero with volume at least . Then for every we have a bound for all x in .

This result can be viewed as a converse to the -noncollapsing property (bounded curvature implies volume noncollapsing). A key point here is that the bound depends only on and not on the -solution itself; this uniformity will be a crucial ingredient in the Perelman compactness theorem below.

Proof. Since is contained in , it suffices to establish the claim when . By replacing r with Ar if necessary we may normalise A=1; we may also rescale . Suppose the claim failed, then there exists a sequence of pointed -solutions with and balls with whose volume is bounded below by for some . Using the point picking argument (Exercise 1 from Lecture 16) we can also ensure that for each r, we have on if n is sufficiently large depending on r. Using the monotonicity and Hamilton’s compactness theorem (Theorem 2 from Lecture 15) we may may thus pass to a subsequence and assume that the flows converge geometrically to a limit , which one easily verifies to be a -solution whose asymptotic volume at time zero is bounded below by . But this contradicts Proposition 3.

– The Perelman compactness theorem –

Corollary 2 leads to another important bound:

Proposition 4 (Bounded curvature at bounded distance). Let , and let be a three-dimensional -solution. Then at time zero, for every and we have on .

Proof. If the claim failed, then there will be an sequence of pointed -solutions and and . Applying Corollary 2 in the contrapositive we conclude that . By the Bishop-Gromov inequality, we can thus find a radius such that (say), where is the volume of the Euclidean 3-ball. By rescaling we may normalise , thus . By Corollary 2 we now have on for every . We may thus use monotonicity and Hamilton compactness as before to extract a limiting solution with and with . But then by the strong maximum principle (see Exercise 7 from Lecture 13), must be flat; since it is -non-collapsed, it must be . But then we have , a contradiction.

Exercise 2. Use Proposition 4 to improve the lim sup in Corollary 1 to a lim inf.

This in turn gives a fundamental compactness theorem.

Theorem 1 (Perelman compactness theorem). Let , and let be a sequence of three-dimensional -solutions, normalised so that . Then after passing to a subsequence, these solutions converge geometrically to another -solution .

Proof. By Proposition 4, we have on for every . Using monotonicity and Hamilton compactness as before, the claim follows.

– Universal noncollapsing –

The Perelman compactness theorem requires to be fixed. However, the theorem can be largely extended to allow for variable by the following proposition.

Proposition 5. (Universal ) There exists a universal such that every 3-dimensional -solution which is not round, is in fact a -solution (no matter how small is).

The reason one needs to exclude the round case is that sphere quotients can be arbitrarily collapsed if one takes to be large (e.g. consider the action of the roots of unity on the unit ball of (which is of course identifiable with ) for n large).

Proof. By time shifting it suffices to show -noncollapsing at time zero at at some spatial origin , which we now fix.

Let be a -solution. By Proposition 1, M is non-compact, which means that any asymptotic gradient shrinking soliton must also be non-compact. By Theorem 1 from the previous lecture, all asymptotic gradient shrinking solitons are thus round shrinking cylinders, or the oriented or unoriented quotient of such a cylinder.

Let be the reduced length function from . Recall from Lecture 15 that one can find a sequence of points with with and on any cylinder , whose rescalings by converge geometrically to an asymptotic gradient shrinking soliton (and thus to a round cylinder or quotient thereof), and the bound does not depend on . A computation shows that these round cylinders or quotients are -noncollapsed for some universal , and so the cylinders are similarly -noncollapsed (for some slightly smaller but universal ). From the bounds on l and R, this implies that reduced volume at time is bounded from below by a constant independent of . Using monotonicity of reduced volume, we thus have this lower bound for all times. The arguments in Lecture 11 then give -noncollapsing for some other universal .

Here is one useful corollary of Perelman compactness and universality:

Corollary 3. (Universal derivative bounds) Let be a three-dimensional -solution. Then we have the pointwise bounds for all . In particular we have .

Proof. The claim is clear for the round shrinking solitons (which we can lift up to live on the sphere ), so we may assume that the -solution is not round. By Proposition 5, we may then replace by a universal . We may then time shift so that t=0 and rescale so that R(0,x)=1. If the claim failed, then we could find a sequence of pointed -solutions with , but such that some derivative of the curvature goes to infinity at this point. But this contradicts Theorem 1.

Here is another useful consequence:

Exercise 3. Let be a sequence of three-dimensional -solutions, and let and . If , show that . (Note that this generalises Corollary 1 or Exercise 2. Hint: the claim is trivial in the round case, so assume non-roundness; then apply universality and compactness.)

– Global structure of -solutions –

Roughly speaking, the above theory tells us that the geometry around any point in a 3-dimensional -solutions has only bounded complexity if we only move in space and in time. This is about as good a control on the local geometry of such solutions as we can hope for; we now turn to the global geometry. [Aside: It is unlikely that the space of 3-dimensional -solutions is finite dimensional, as it is in the 2-dimensional case; see for instance Example 1.4 of Perelman's second paper for what is probably an infinite-dimensional family of -solutions.]

Let us begin with non-compact 3-dimensional -solutions. A key point is that if such solutions are not already round cylinders (or quotients thereof), they must mostly resemble such cylinders.

Definition 1. (Necks) Let . An -neck in a Riemannian 3-manifold centred at a point is a diffeomorphism from a long cylinder to M, such that the normalised pullback metric lies within of the standard round metric on the cylinder in the topology, where we require of course that R(x) > 0. The number is called the width scale of the neck, and is the length scale.

Clearly, the notion of a -neck is a scale-invariant concept. Note that if a sequence of pointed manifolds is converging geometrically (after rescaling) to a round cylinder , then for any , will be in the centre of an -neck for sufficiently large n. Since round cylinders appear prominently as geometric limits, it is then not surprising that -solutions, particularly non-compact ones, tend to be awash in -necks. For instance, we have

Proposition 6. For every there exists an such that whenever is a point in a 3-dimensional non-compact -solution of strictly positive curvature and is a unit speed minimising geodesic from x to infinity (such things can easily be shown to exist by compactness arguments) at time t, then every point in lies in the centre of an -neck at time t.

Proof. By time shifting we can take t=0. Suppose the claim is not the case, then we have a sequence of pointed 3-dimensional non-compact -solutions of strictly positive curvature and on a minimising geodesic from to infinity such that and is not the centre of a -neck at time zero. By Exercise 3 we thus have . Let us now rescale so that . Since the are non-compact, they are non-round and so by Proposition 5 we can take to be universal, at which point by Perelman compactness (Theorem 1) we can pass to a subsequence and assume that is converging to a limit , which is also a -solution. Since , we see that the limit manifold contains a minimising geodesic line through , and hence by the Cheeger-Gromoll splitting theorem (Theorem 2 from Lecture 16) must split into the product of a line and a positively curved manifold. By Proposition 2, we conclude that is either a cylinder or a projective cylinder .

The latter can be ruled out by topological considerations; a positively curved complete non-compact 3-manifold is homoemorphic to by the soul theorem, and so does not contain any embedded with trivial normal bundle. (In any event, for applications to the Poincaré conjecture one can always assume that no such embedded projective plane exists in any manifold being studied.) So is a round cylinder, and thus is the centre of an -neck, a contradiction, and the claim follows.

There is a variant of Proposition 6 that works in the compact case also:

Proposition 7. For every there exists an such that whenever are points in a 3-dimensional -solution (either compact or noncompact) then at time t, any point on the minimising geodesic between x and y at a distance at least from x and from y, lies in the centre of an -neck at time t.

Proof. We can repeat the proof of Proposition 6. The one non-trivial task is the topological one, namely to show that M does not contain an embedded with trivial normal bundle in the compact case (the non-compact case already being covered in Proposition 6). But M is compact and has strictly positive curvature (thanks to Proposition 2) and so by Hamilton’s rounding theorem, is diffeomorphic to a spherical space form for some finite ; in particular the fundamental group is finite. On the other hand, an embedded with trivial normal bundle cannot separate M (as its Euler characteristic is 1) and so a closed loop in M can have a non-trivial intersection number with such a projective plane (using the normal bundle to give a sign to each intersection), leading to a non-trivial homomorphism from to , contradicting the finiteness of the fundamental group. [An alternate argument would be to use Perelman compactness to extract a non-compact (but positively curved) limiting -solution from a sequence of increasingly long compact -solutions. Proposition 6 prohibits the limiting solutions from asymptotically looking like , and so the long compact solutions cannot have such projective necks either.]

Informally, the above proposition shows that any two sufficiently far apart points in a compact -solution will be separated almost entirely by -necks. Since the only way that necks can be glued together is by forming a tube, one can then show the following two corollaries:

Corollary 4. (Description of non-compact positively curved -solutions) For every there exists such that for every non-compact 3-dimensional positively curved -solution and time t there exists a point such that at time t

1. Every point outside of lies in an -neck (and in particular, the exterior of this ball is topologically a half-infinite cylinder ); and
2. Inside the ball (which is topologically a standard 3-ball by the soul theorem) all sectional curvatures are comparable to R(p) modulo constants C depending only on , and the volume of the ball is comparable to modulo similar constants C.

(The control inside the ball is coming from results such as Corollary 3, as well as the non-collapsed nature of M.)

In the language of Morgan-Tian, we have described non-compact positively curved 3-dimensional -solutions as C-capped -tubes. [Actually, Morgan-Tian prove a little more: they control the time evolution of the necks and not just individual time slices, leading to the notion of a strong -neck. See Section 9.8 of that book for details, as well as a precise definition of the C-capped -tubes.] Combined with Proposition 2, we now have a satisfactory description of non-compact -solutions: they are either round cylinders (and thus doubly infinite -tubes), oriented quotients of round cylinders (and thus a half-infinite -tube capped off by a punctured ), oriented quotients of round cylinders (and thus containing an with trivial normal bundle), or a half-infinite -tube capped off by a 3-ball.

For compact -solutions, we have something similar:

Proposition 8. (Characterisation of large compact -solutions) For every there exists such that if is a compact 3-dimensional -solution with at some time t, then can be partitioned into an -tube (roughly speaking, a region in which every point lies in the middle of an -neck, and bordered on both ends by an ) and two -caps (roughly speaking, two regions diffeomorphic to either a 3-ball or punctured , bounded by an , in which the sectional curvatures are comparable to a scalar R, the diameter is comparable to , and volume comparable to ). See Section 9.8 of Morgan-Tian for precise definitions.

The topological characterisation of the caps (that they are either 3-balls or punctured s) follows from the corresponding characterisations of the caps in the non-compact case, followed by a compactness argument. Note that the round compact manifolds have diameter , where R is the constant curvature, and thus are not covered by the above Proposition.

By considering the various topologies for the caps, we see from basic topology then tells us that the manifolds in this case are homeomorphic to either or , or . The latter has infinite fundamental group, though, and thus not homeomorphic to a spherical space form; thus it cannot actually arise since Hamilton’s rounding theorem asserts that all compact manifolds of positive curvature are homeomorphic to spherical space forms.

Finally, we turn to small compact non-round -solutions.

Proposition 9. (Characterisation of small compact -solutions) Let , and let be a compact 3-dimensional -solution with at some time t which is not round, then all sectional curvatures are comparable up to constants depending on C, the diameter is comparable to up to similar constants, the volume is comparable to , and the manifold is topologically either or .

Proof. The diameter, curvature, and volume bounds follow from the compactness theory. To get the topological type, observe from the treatment of the compact soliton case that as M is not round, the asymptotic gradient shrinking soliton is non-compact, and thus must be a cylinder or one of its quotients. In particular this implies that as one goes back in time, the manifold M must eventually become large in the sense of Proposition 8. Since the manifolds in that proposition were topologically either or , the same is true here.

Putting all of the above results together, we obtain Proposition 1 from Lecture 12 (modulo some imprecision in the definitions which I have decided not to detail here).

[Updated, June 3: Proposition 9 added.]

1. It is ancient: t ranges over .
2. It is a Ricci flow.
3. M is complete and connected.
4. The Riemann curvature is non-negative (though it could theoretically be unbounded).
5. is non-negative.
6. M is -noncollapsed.
7. M is not flat.
8. It obeys the gradient shrinking soliton equation

(1)

for some smooth f.

The main result of this lecture is to classify all such solutions in low dimension:

Theorem 1. (Classification of asymptotic gradient shrinking solitons) Let be as above, and suppose that the dimension d is at most 3. Then one of the following is true (up to isometry and rescaling):

1. d=2,3 and M is a round shrinking spherical space form (i.e. a round shrinking , , , or for some finite group acting freely on ).
2. d=3 and M is the round shrinking cylinder or the oriented or unoriented quotient of this cylinder by an involution.

The case d=2 of this theorem is due to Hamilton; the compact d=3 case is due to Ivey; and the full d=3 case was sketched out by Perelman. In higher dimension, partial results towards the full classification (and also relaxing many of the hypotheses 1-8) have been established by Petersen-Wylie, by Ni-Wallach, and by Naber; these papers also give alternate proofs of Perelman’s classification.

To prove this theorem, we induct on dimension. In 1 dimension, all manifolds are flat and so the claim is trivial. We will thus take d=2 or d=3, and assume that the result has already been established for dimension d-1. We will then split into several cases:

1. Case 1: Ricci curvature has a zero eigenvector at some point. In this case we can use Hamilton’s splitting theorem to reduce the dimension by one, at which point we can use the induction hypothesis.
2. Case 2: Manifold noncompact, and Ricci curvature is positive and unbounded. In this case we can take a further geometric limit (using some Toponogov theory on the asymptotics of rays in a positively curved manifold) which is a round cylinder (or quotient thereof), and also a gradient steady soliton. One can easily rule out such an object by studying the potential function of that soliton on a closed loop.
3. Case 3: Manifold noncompact, and Ricci curvature is positive and bounded. Here we shall follow the gradient curves of f using some identities arising from the gradient shrinking soliton equation to get a contradiction.
4. Case 4: Manifold compact, and curvature positive. Here we shall use Hamilton’s rounding theorem to show that one is a round shrinking sphere or spherical space form.

We will follow Morgan-Tian‘s treatment of Perelman’s argument; see also the notes of Kleiner-Lott, the paper of Cao-Zhu, and the book of Chow-Lu-Ni for other treatments of this argument.

– Case 1: Ricci curvature degenerates at some point –

This case cannot happen in two dimensions. Indeed, since the Ricci curvature is conformal in this case, the only way that the Ricci curvature can degenerate is if the scalar curvature vanishes also. But then the strong maximum principle (Exercise 7 from Lecture 13) forces the gradient shrinking soliton to be flat at all sufficiently early times (and hence at all times), a contradiction. (It turns out that this application of strong maximum principle can be extended to cover the case in which one does not have bounded curvature.)

So now suppose that we are in three dimensions with bounded Ricci curvature, and a point where the Ricci curvature vanishes. Then by Hamilton’s splitting theorem (Proposition 1 from Lecture 13) the gradient shrinking soliton locally splits into the product of a two-dimensional flow and a line (for sufficiently early times, at least), with the Ricci curvature being degenerate along these lines that foliate the flow. (Again, one has to extend the strong maximum principle argument to cover the case of unbounded curvature, but this can be done.) In particular, from (1) we see that is constant and strictly positive along these lines; in other words, f is strictly convex (and quadratic) along these lines. As a consequence, the lines cannot loop back upon themselves.

By lifting to a double cover if necessary, we can find a global unit vector field X along these lines, thus and . If we set , we conclude from (1) that , thus the level sets of F have X as a unit normal. Thus, at any fixed time, we use F to globally split the manifold M (or a double cover thereof) as the product of a line and a two-dimensional manifold (given by the level sets of F). Applying the induction hypothesis, we conclude that M (or a double cover) is a product of a line and a round shrinking or (as these are the only two-dimensional spherical space forms), at which point we end up in alternative 2 of Theorem 1. (We initially establish this fact only for sufficiently early times, but then by uniqueness of Ricci flow one obtains it for late times also.)

Remark 1. We can also proceed here using the global splitting theorem from Lemma 9.1 of Hamilton’s paper.

– Case 2: Manifold non-compact, curvature positive and unbounded –

Now we handle the case in which M is non-compact (and in particular has a meaningful notion of convergence to spatial infinity) with Ricci curvature strictly positive and unbounded. In particular one has a sequence of points in M such that

(2)

at some time (which we can normalise to t=-1), where we arbitrarily pick an origin . Thus the curvature is not decaying as fast as at infinity, and may even be unbounded. Henceforth we normalise t as t=-1 and write g for g(-1).

The basic idea here is to look at the rescaled pointed manifolds and extract a limit in which the original base point has now been sent off to infinity (thanks to (2)). There is a technical obstacle to doing this, though, which is that the rescaled manifolds have bounded curvature at (indeed, it has been normalised to equal 1) but might have unbounded curvature at nearby points with respect to the rescaled metric (i.e. points within distance in the original metric) because such points may have significantly higher curvature than (e.g. ). But it is easy to resolve this: simply pick instead of . Now may itself be close to another point of even higher curvature, but we can then move that point instead. We can continue in this manner, moving in a geometrically decreasing sequence of distances, until we stop (which we must, since the manifold is smooth and so curvature is locally bounded). The precise result of this “point-picking argument”, originally due to Hamilton, that we will need is as follows:

Exercise 1. (Point picking lemma) Assuming that (2) holds for some sequence , show that there exists another sequence also obeying (2), and such that for any , and for all n sufficiently large depending on A, we have for all . If the original manifold had unbounded curvature, show that we can also ensure that .

We now let be as above, and consider the rescaled manifolds . Using Hamilton’s compactness theorem (Theorem 2 from Lecture 15) we may assume that these manifolds converge geometrically to a limit of nonnegative Riemann curvature whose scalar curvature is at most 4 (and is equal to 1 at ); in particular the limit has bounded curvature. From the analogue of (2) for we have , and so has “escaped to infinity” in the limit (this shows in particular that is non-compact).

Let , thus . By refining this sequence we may assume that we have rapid growth in the sense that . Let be a minimising geodesic from to ; by compactness we may assume that the direction of at is convergent. In particular, the angle subtended between and is . If we let be a minimising geodesic from to , we thus see from the triangle inequality and the cosine rule (Lemma 2) that

. (3)

Using the cosine rule again, we see that the angle subtended between and is . Using relative Toponogov comparison (Exercise 3) we see that the rays and asymptotically form a minimising geodesic, in the sense that for any z, w at a bounded distance away from on and respectively. From this, we see in the limit that there exists a minimising geodesic line through . But by the Cheeger-Gromoll splitting theorem (Theorem 2) we see that splits into the product of a line and a manifold of one dimension less. This cannot happen in the two-dimensional case d=2, since becomes one-dimensional and thus flat, and has non-zero curvature at (indeed, its scalar curvature is equal to 1). So we can now assume d=3.

We have only taken limits at time t=-1. But we can use Hamilton’s compactness theorem (Theorem 2 from Lecture 15) again (using the property ) and extend to a Ricci flow backwards in time from t=-1; this is a limit of rescaled versions of by . Since M was originally a gradient shrinking soliton, and is going to infinity, the limit can be shown to be a gradient steady soliton: for some .

Since had bounded curvature at time t=-1, it had bounded curvature for all previous times also. Since the Ricci curvature is vanishing along one direction, we can now apply the Case 1 argument and show that is the product of a line and a round shrinking or . In particular, contains closed geodesic loops on which the Ricci curvature is strictly positive. From the gradient steady equation, this means that is strictly concave on this loop, which is absurd. Thus this situation does not occur.

Remark 2. In Morgan-Tian, the contradiction was obtained using the soul theorem, and a rather non-trivial result asserting that complete manifolds of non-negative sectional curvature cannot contain arbitrarily small necks, but the above argument seems to be somewhat shorter.  An even simpler argument (avoiding the use of the splitting theorem altogether) was given by Naber, based on the observation (from (1)) that the normalised gradient vector field of the potential function becomes increasingly parallel to the connection if goes to infinity.  We thank Peter Petersen for pointing out Naber’s argument to us.

– Case 3: M noncompact, curvature positive and bounded –

Now we assume that M is compact, with Ricci curvature strictly positive but also bounded. By Lemma 1 of Lecture 15, we conclude in particular that

(4)

for some C and all minimising geodesics (thus the Ricci curvature must decay along long geodesics). On the other hand, along such a geodesic, we see from (1) that

. (5)

From (4) and (5) we see that increases like as . Similarly, if E is any vector field orthogonal to X and transported by parallel transport along , an application of Cauchy-Schwarz, (4), and the bounded curvature hypothesis gives

(6)

while (1) gives

(7)

and so grows like at most as . These bounds ensure that f goes to at infinity (in particular, it is proper), and that there exist curves following the gradient of f which go to infinity.

On the other hand, using the identity

(8)

(see (27) from Lecture 13) we see that , thus R is increasing along gradient flow curves. In particular, is strictly positive (and finite, since curvature is bounded).

As a consequence, we can repeat the point-picking arguments from Case 2 and extract a sequence of points for which converges geometrically to a limit , which has scalar curvature at . Since M is a gradient shrinking soliton on , one can show that is also. By repeating the Case 2 analysis one can show that is also a round shrinking or . Since these solitons have scalar curvature 1 at time -1, we thus have .

For sake of argument let us take M to be the round shrinking cylinder ; the other case is similar but with all areas divided by a factor of two. (One can also eliminate this case by appealing to the soul theorem, or by adding an additional hypothesis throughout the argument that the manifolds being studied do not contain embedded ‘s with trivial normal bundle.)

Now we return to the original gradient shrinking soliton M. Since R is strictly increasing along gradient flow curves, we conclude that near infinity. Since M has non-negative Riemann curvature, this implies near infinity. From (1) this implies that f is strictly convex (i.e. ) near infinity. Thus the level sets of f have increasing area. On the other hand, on any region of M that approaches (e.g. in the neighbourhoods of ) one easily sees (e.g. from (1), or from the analysis from Case 2) that the level sets of f converge to the sections of the cylinder, which have area (note we are normalising the scalar curvature here to be 1, rather than the sectional curvature, which is 1/2). Thus the level sets of f have area strictly less than .

On the other hand, from the Gauss-Codazzi formula (equation (4) from Lecture 4), the Gaussian curvature K of is given by the formula

(9)

where is the sectional curvature of , and is the second fundamental form. Applying (1) we eventually compute

. (10)

Following the gradient flow lines of f, we see from previous analysis that goes to infinity (while curvature stays bounded and strictly positive), and so it is not hard to see that the right-hand side must be strictly less than 1 near infinity. But this means that , contradicting the Gauss-Bonnet formula (Proposition 1 from Lecture 4). Thus Case 3 cannot in fact occur.

– Case 4: M compact, strictly positive curvature –

Let us first deal with the two-dimensional case. Here one could use Hamilton’s results on Ricci flow for surfaces to show that this gradient shrinking soliton must be a round shrinking or , but we give here an argument adapted from the book of Chow and Knopf. It relies on the following identity, that provides an additional global constraint on the curvature R beyond that provided by the Gauss-Bonnet theorem:

Lemma 1 (Kazhdan-Warner type identity) Let (M,g) be a compact surface, and let X be a conformal Killing vector field (thus is a scalar multiple of g). Then .

Proof. When M has constant curvature, the claim is clear by integration by parts. On the other hand, by the uniformization theorem, any metric g can be conformally deformed to a constant curvature metric. Note also from definition that a conformal Killing vector field remains conformal after any conformal change of metric. Thus it suffices to show that is constant under any conformal change of g, keeping X static.

From the variation formulae from Lecture 1, we have and . Inserting these formulae and integrating by parts to isolate u, we see that it suffices to show that . On the other hand, since is conformal, we have the identity . Taking divergences of this identity twice and rearranging derivatives repeatedly, we eventually obtain this claim.

[Aside: I do not know of any proof of the Kazhdan-Warner identity that does not require the uniformisation theorem; the result seems to have an irreducibly "global" nature to it.]

Now we apply this lemma to the vector field , which is conformal thanks to (1). We conclude that . On the other hand, from the trace of (1) we have . Integrating this against we conclude that , thus f is harmonic; and so . M is now constant curvature and is therefore either a round shrinking or as required.

Now we turn to three dimensions. The result in this case follows immediately from Hamilton’s rounding theorem, but we will take advantage of the gradient shrinking soliton structure to extract just the key components of that theorem here. Let denote the eigenvalues of the Riemann curvature. Note that as the Ricci curvature is positive, is strictly greater than zero.

The quantity ranges between 0 and 2 and reaches a minimum value at some point x. If we rewrite things in terms of the tensor from Lecture 3, the gradient shrinking soliton structure means that

(11)

But the region is fibrewise convex and parallel, and at x, and are tangential to this region and is tangential or inward. On the other hand, a computation shows that is strictly inward unless , in which case it is tangential. So we must have , which implies that . In other words, the Ricci tensor is conformal: . Comparing this with the Bianchi identity (equation (28) from Lecture 0) we conclude that , and thus . Thus M has constant sectional curvature and is therefore a round shrinking spherical space form, as required.

– Appendix: Toponogov theory –

Roughly speaking, Toponogov comparison theory is to triangle geometry as Bishop-Gromov theory is to volumes of balls: in both cases, lower bounds on curvature are used to bound the geometry of Riemannian manifolds by model geometries such as Euclidean space. This theory links modern Riemannian geometry with the more classical approach to curved space (or non-Euclidean geometries) which often proceeded via analysing the angles formed by a triangle. The material here is loosely drawn from Petersen’s book.

Lemma 2. (Toponogov cosine rule) Let be a complete Riemannian manifold of non-negative sectional curvature, and let be three distinct points in M. Let be the angle formed at by the minimising geodesics from to . Then

(12).

Of course, when M is flat we have equality in (12), by the classical cosine rule.

Proof. Let f be the function , and let be the unit speed geodesic from to . Our task is to show that

(13)

for t = d(x_2,x_1). From the Gauss lemma we know that . On the other hand, from the second variation formula for distance (or more precisely, equation (21) of Lecture 10) and the non-negative sectional curvature assumption we have . (Actually one has to justify this in a suitable barrier sense when one is in the cut locus, but let us ignore this issue here for simplicity.) The claim follows.

There is an appealing reformulation of this lemma. Define a triangle to be three points A, B, C connected by three minimising geodesics AB, BC, CA.

Exercise 2. (Positive curvature increases angles) Let ABC be a triangle in a Riemannian manifold of non-negative sectional curvature, and let A’B'C’ be a triangle in Euclidean space with the same side lengths as ABC. Show that the angle subtended at A is larger than or equal to that subtended at A’ (and similarly of course for B and B’, and C and C’). In particular, the sum of the angles of ABC is at least .

There is also a relative version of this result:

Exercise 3. (Relative Toponogov comparison) Let the notation and assumptions be as in the previous exercise. Let X, Y be points on AB, AC respectively, and let X’, Y’ be the corresponding points on A’B’ and A’C’. Show that the length of XY is greater than or equal to the length of X’Y’. (Hint: it suffices to do this in the case X=B (or Y=C), since the general case follows by two applications of this special case. Now repeat the argument used to prove Lemma 2.)

Remark 3. Similar statements hold when one assumes that the sectional curvatures are bounded below by some number K other than zero. In this case, one replaces Euclidean space with the model geometry of constant curvature K, much as in the discussion of the Bishop-Gromov inequality in Lecture 9. See Petersen’s book for details.

– The Cheeger-Gromoll splitting theorem –

When a manifold has positive curvature, it is difficult for long geodesics to be minimising; see for example Myers’ theorem for one instance of this phenomenon.
Another important example of this is the Cheeger-Gromoll splitting theorem.

Theorem 2 (splitting theorem). Let be a complete Riemannian manifold of nonnegative Ricci curvature that contains a minimising geodesic line . Then M splits as the product of with a manifold of one lower dimension.

Remark 4. If one strengthens the non-negative Ricci curvature assumption to non-negative sectional curvature, this is a result of Toponogov; if one strengthens further to have a uniform positive lower bound on sectional curvature, then this follows from Myers’ theorem.

Proof. We can parameterise to be unit speed. Consider the Busemann functions defined by

(13)

One can show that the limits exist (because, by the triangle inequality, the expressions in the limits are bounded and monotone), and that are both Lipschitz. From the non-negative curvature we have the upper bound for any distance function (see e.g. equation (21) from Lecture 10); applying this with and letting we obtain the concavity . In particular, is concave. On the other hand, from the triangle inequality we see that is non-negative and vanishes on . Applying the (elliptic) strong maximum principle (which can be viewed as the static case of the parabolic strong maximum principle, Exercise 5 from Lecture 13, though in the static case the bounded curvature hypothesis is not needed) we conclude that vanishes identically. Since and were both concave, they now must flat in the sense that . In particular they are smooth, and the gradient vector field is parallel to the Levi-Civita connection. On the other hand, by applying the Gauss lemma carefully we see that X is a unit vector field. Thus X splits M into a line and the level sets of (cf. Proposition 1 from Lecture 13) as desired.

Next, we use the estimates on reduced length from the Harnack inequality analysis in Lecture 13 to locate some good regions of spacetime of a -solution in which to do the asymptotic analysis. Rescaling these regions and applying Hamilton’s compactness theorem (relying heavily here on the -noncollapsed nature of such solutions) we extract a limit. Formally, the reduced volume is now constant and so Lecture 14 suggests that this limit is a gradient soliton; however, some care is required to make this argument rigorous. In the next section we shall study such solitons, which will then reveal important information about the original -solution.

Our treatment here is primarily based on Morgan-Tian’s book and the notes of Ye. Other treatments can be found in Perelman’s original paper, the notes of Kleiner-Lott, and the paper of Cao-Zhu. See also the foundational papers of Shi and Hamilton, as well as the book of Chow, Lu, and Ni.

– Geometric limits –

To develop the theory of geometric limits for pointed Ricci flows , we begin by studying such limits in the simpler context of pointed Riemannian manifolds , i.e. a Riemannian manifold together with a point , which we shall call the origin or distinguished point of the manifold. To simplify the discussion, let us restrict attention to complete Riemannian manifolds (though for later analysis we will eventually have to deal with incomplete manifolds).

Definition 1. (Geometric limits) A sequence of pointed d-dimensional connected complete Riemannian manifolds is said to converge geometrically to another pointed d-dimensional connected complete Riemannian manifold if there exists a sequence of connected neighbourhoods of increasing to (i.e. ) and a sequence of smooth embeddings mapping to such that

1. The closure of each is compact and contained in (note that this implies that every compact subset of will be contained in for sufficiently large n);
2. The pullback metric converges in the topology to (i.e. all derivatives of the metric converge uniformly on compact sets).

Example 1. The pointed round d-sphere of radius R converges geometrically to the pointed Euclidean space as . Note how this example shows that the geometric limit of compact manifolds can be non-compact.

Example 2. If (M,g) is Hamilton’s cigar (Example 3 from Lecture 8), and is a sequence on M tending to infinity, then converges geometrically to the pointed round 2-cylinder.

Example 3. The d-torus of length 1/n does not converge to a geometric limit as , despite being flat. More generally, the sequence needs to be locally uniformly non-collapsed in order to have a geometric limit.

Exercise 1. Show that the geometric limit of a sequence , if it exists, is unique up to (pointed) isometry.

Geometric limits, as their name suggests, tend to preserve all (local) “geometric” or “intrinsic” information about the manifold, although global information of this type can be lost. Here is a typical example:

Exercise 2. Suppose that converges geometrically to . Show that for every , and that we have the Fatou-type inequality . Give an example to show that the latter inequality can be strict.

Here is the basic compactness theorem for such limits.

Theorem 1. (Compactness theorem) Let be a sequence of connected complete Riemannian d-dimensional manifolds. Assume that

1. (Uniform bounds on curvature and derivatives) For all , one has the pointwise bound on the ball for all sufficiently large n and some constant .
2. (Uniform non-collapsing) For every there exists such that for all and , and all sufficiently large n.

Then, after passing to a subsequence if necessary, the sequence has a geometric limit.

Proof. (Sketch) Let be an arbitrary radius. From Cheeger’s lemma (Theorem 1 from Lecture 7) and hypothesis 2, we know that the injectivity radius on is bounded from below by some small for sufficiently large n. Also, from the curvature bounds and Bishop-Gromov comparison geometry (Lemma 1 from Lecture 9) we know that the volume of is uniformly bounded from above for sufficiently large n.

Now find a maximal -net of , thus the balls are disjoint and the balls cover . Volume counting shows that k is bounded for all sufficiently large n; by passing to a subsequence we may assume that it is constant. Similarly we may assume that all the distances converge to a limit. Using the exponential map and some arbitrary identification of tangent spaces with , we can identify each ball with the standard Euclidean ball of radius . Any pair of separation less than induces a smooth transition map from the Euclidean ball of radius into some subset of , which can be shown by comparison geometry to be uniformly bounded in norms; applying the ( version of the) Arzelá-Ascoli theorem we may thus pass to a subsequence and assume that all these transition maps converge in to a limit. It is then a routine matter to glue together all the limit transition maps to fashion an incomplete manifold to which the balls converge geometrically (up to errors of at the boundary). Furthermore, as one increases , one can show (by a modification of Exercise 1) that these limits are compatible. Now letting go to infinity (and using the usual diagonalisation trick on all the subsequences obtained), and then gluing together all the incomplete limits obtained, one can create the full geometric limit.

Remark 1. One could use ultrafilters here in place of subsequences, but this does not significantly affect any of the arguments.

Now we turn to geometric limits of pointed Ricci flows (Ricci flows with a specified origin ).

Definition 2. Let be a sequence of pointed d-dimensional complete connected Ricci flows, each on its own time interval . We say that a pointed d-dimensional complete connected Ricci flow on a time interval I is a geometric limit of this sequence if

1. Every compact subinterval of I is contained in for all sufficiently large n.
2. There exists neighbourhoods of as in Definition 1, compact time intervals increasing to I, and smooth embeddings preserving the origin such that the pullback of the flow to converges in spacetime to .

Exercise 3. Show that if a sequence of -noncollapsed Ricci flows (with a uniform value of ) converges geometrically to another Ricci flow, then the limit flow is also -noncollapsed.

Now we present Hamilton’s compactness theorem for Ricci flows, which requires less regularity hypotheses than Theorem 1 due to the parabolic smoothing effects of Ricci flow (as captured by Shi’s estimates, see Theorem 3).

Theorem 2. (Hamilton compactness theorem) Let and be as in Definition 2, and let I be an open interval obeying hypothesis 1 of that definition. Let be a time. Suppose that

1. For every compact subinterval J of I containing and every , one has the curvature bound on the cylinder for some and all sufficiently large n; and
2. One has the non-collapsing bound for some and , and all sufficiently large n.
3. (Uniform lower bound on curvature)  For any compact J, there is a K such that for any , one has the curvature lower bound on for all sufficiently large n.  (This is not quite implied by 1. because the curvature bound K there is allowed to depend on r, whereas here we require that K is independent of r.)

Then some subsequence of converges geometrically to a limit on I.

Condition 3 is technical (and was erroneously omitted in some of the literature), but it was recently observed by Topping that the claim fails without some hypothesis of this form.  Fortunately, in the applications to the Poincare conjecture one always has a uniform lower bound on Ricci curvature, and so Condition 3 is not difficult to verify in practice.

Proof. By Shi’s estimates (Theorem 3) we can upgrade the bound on curvature in hypothesis 1 to bounds on derivatives of curvature. Indeed, these estimates imply that for any J, r as in that hypothesis, and any , we have for some and sufficiently large n.

Now we restrict to the time slice and apply Theorem 1. Passing to a subsequence, we can assume that converges geometrically to a limit .

For any radius r and any compact J in I containing , we can pull back the flow to a (spatially incomplete) flow on the cylinder for sufficiently large n. By construction, converges in norm to ; in particular, it is uniformly bounded in each of the seminorms of this space. Also, each is a Ricci flow with uniform bounds on any derivative of curvature for sufficiently large n.

Exercise 4. Using these facts, show that the sequence of flows is uniformly bounded in each of the seminorms of for each fixed J, r, and for n sufficiently large.

By using the Arzelá-Ascoli theorem as before, we may thus pass to a further subsequence and assume that converges in to a limiting flow . Clearly this limit is a Ricci flow. Letting and pasting together the resulting limits one obtains the desired geometric limit. (One has to verify that every geodesic in starting from can be extended to any desired length, thus establishing completeness by the Hopf-Rinow theorem, but this is easy to establish given all the uniform bounds on the metric and curvature, and their derivatives. It is here that hypothesis 3 is used to prevent the metric from shrinking too rapidly as one goes backwards in time. )

– Locating an asymptotic gradient shrinking soliton –

We now return to the study of -solutions . We pick an arbitrary point and consider the reduced length function . Recall (see equation (18) from Lecture 11) that we had

(1)

for every . (This bound was obtained from the parabolic inequality and the maximum principle.) Thus we can find a sequence with such that

. (2)

Now recall that as a consequence of Hamilton’s Harnack inequality, we have the pointwise estimates

(3)

and

(4)

(see equations (37), (38) from Lecture 13). From these bounds and Gronwall’s inequality, one easily sees that we can extend (2) to say that

(5)

for any in the cylinder and any and . Applying (3) once more, together with the hypothesis of non-negative curvature more, we also obtain bounded normalised curvature on this cylinder:

(6).

If we thus introduce the rescaled flow by setting , , and , we see that these flows obey hypothesis 1 of Theorem 2. Also, since the original -solutions are -noncollapsed, so are their rescalings, which (in conjunction with hypothesis 1) gives us hypothesis 2. We can thus invoke Theorem 2 and assume (after passing to a subsequence) that the rescaled flows converge geometrically to an ancient Ricci flow on the time interval . From Exercise 3 we see that this limit is also -noncollapsed. Since the rescaled flows have non-negative curvature, the limit flow has non-negative curvature also. (Note however that we do not expect in general that has bounded curvature (for instance, if the original -solution was a round shrinking sphere terminating at the unit radius sphere, the limit object would be a round shrinking sphere terminating at a point). In particular we do not expect to be a -solution.)

Let be the rescaled length function, thus . From (5) we see that is uniformly bounded on compact subsets of for n sufficiently large (where we identify compact subsets of with subsets of for n large enough). By the rescaled versions of (4) and (5) we also see that is also uniformly bounded on such compact sets for sufficiently large n; thus the are uniformly Lipschitz on each compact set. Applying the Arzelá-Ascoli theorem and passing to a subsequence, we may thus assume that the converge uniformly on compact sets to some limit , which is then locally Lipschitz.

Remark 2. We do not attempt to interpret as a reduced length function arising from some point at time t=0; indeed we expect the limiting flow to develop a singularity at this time.

We know that the reduced volume is non-increasing in and ranges between 0 and , and so converges to a limit between 0 and . This limit cannot equal since this would mean that the -solution is flat (by Theorem 1 from Lecture 14), which is absurd. The limit cannot be zero either, since the bounds (5) and the non-collapsing ensure a uniform lower bound on the reduced volume. By rescaling, we conclude that

(7)

for each fixed .

Let us now argue informally, and then return to make the argument rigorous later. Formally taking limits in (7), we conclude that

. (8)

On the other hand, from the proof of the monotonicity of reduced volume from Lecture 10 we have (formally, at least)

(9)

and hence by rescaling

. (10)

Formally taking limits, we obtain

. (11)

We can rewrite this as the assertion that is a subsolution of the backwards heat equation:

. (12)

This (formally) implies that the left-hand side of (8) is non-increasing in . On the other hand, this quantity is constant in ; and so (12) must be obeyed with equality, and thus

. (13)

Also, recall from Lecture 10 that

. (14)

Rescaling and taking limits, we formally conclude that the same is true for ;

. (15)

From (14) and (15) we obtain that the Perelman -functional

(16)

vanishes (cf. the last section of Lecture 11). In particular, it is constant. On the other hand, by (13) and the monotonicity formula for this functional (see Exercise 9 of Lecture 8) we have

. (17)

Combining this with the vanishing of (16) we thus conclude that

(18)

and thus is a gradient shrinking soliton as desired.

– Making the argument rigorous I. Spatial localisation –

Now we turn to the (surprisingly delicate) task of justifying the steps from (7) to (18).

The first task is to deduce (8) from (7). From the dominated convergence theorem it is not difficult to show that

(19)

for any fixed and r; the difficulty is to prevent the escape of mass of to spatial infinity. (Fatou’s lemma will tell us that the left-hand side of (8) is less than or equal to the right, but this is not enough for our application.)

In order to prevent such an escape, one needs a lower bound on when is large. (Note that estimates such as (3), (4) only provide upper bounds on .) The problem is equivalent to that of upper bounding in terms of . To do this we need some control on quantities related to the distance function at extremely large distances. Remarkably, such bounds are possible. We begin with a lemma of Perelman (related to an earlier argument of Hamilton).

Lemma 1. Let be a d-dimensional Riemannian manifold, let , and let . Suppose that on the balls and . Then for any minimising geodesic connecting x and y, we have , where is the velocity field.

Proof. We may assume that , since the claim is trivial otherwise. We recall the second variation formula

(20)

whenever one deforms a geodesic along a vector field Y (see equation (17) of Lecture 10). Since is minimising, the left-hand side of (20) is non-negative when Y vanishes at the endpoints of . Now let v be any unit vector at x, transported by parallel transport along . Setting Y(t) to equal tv/r when , equal to v when , and equal to (d(x,y)-t)v/r when , we conclude that

. (21)

Letting v vary over an orthonormal frame and summing, we soon obtain the claim.

The above lemma, combined with the Ricci flow equation, gives an upper bound as to how rapidly the distance function can grow as one goes backwards in time.

Corollary 1. Let be a d-dimensional Ricci flow, let , let t be a time, and let . Suppose that on and on . Then (in the sense of forward difference quotients).

Using this estimate, we can now obtain a bound on distance in terms of reduced length.

Proposition 1. Let be a d-dimensional -solution, let , and . Then

. (22)

Proof. We use an argument of Ye. Write A for the expression inside the on the right-hand side, and let be minimising -geodesics from to p, p’ respectively. By the fundamental theorem of calculus, we have

. (24)

Using (3) and the -Gauss lemma we see that move at speed , and that all curvature tensors are in a -neighbourhood of either curve. Applying Corollary 1, the chain rule, and the Gauss lemma we conclude that

; (25)

inserting this into (24) we obtain the claim.

Combining this with (5) and rescaling we see that we have a bound of the form

(26)

for all x and some ; taking limits we also obtain

. (27)

On the other hand, from the Bishop-Gromov inequality we know that balls of radius r in either or have volume . These facts are enough to establish that the portion of (7) or (8) outside of the ball of radius r decays exponentially fast in r, uniformly in n, and this allows us to take limits in (19) as to deduce (8) from (7).

– Making the argument rigorous II. Parabolic inequality for –

The next major task in making the previous arguments rigorous is to justify the passage from (10) to (11). First of all, because of the -cut locus, (10) is only valid in the sense of distributions. We would like to take limits and conclude that (11) holds in the sense of distributions as well. There is no difficulty taking limits with the linear terms in (10), or in the zeroth order terms ; the only problem is in justifying the limit from to . We know that the are uniformly locally Lipschitz, and converge locally uniformly to ; but this is unfortunately not enough to ensure that converges in the sense of distributions to , due to possible high frequency oscillations in . To give a toy counterexample, the one-dimensional functions are uniformly Lipschitz and converge uniformly to zero, but converges in the distributional sense to rather than zero.

Since is bounded and converges distributionally to zero, it will be locally asymptotically orthogonal . From this and Pythagoras’ theorem we obtain

(28)

in the sense of distributions, where we pass to a subsequence in order to make the limits on both sides exist. (Note that converges locally uniformly to g and so there is no difficulty passing back and forth between those metrics.) The task is now to show that there is not enough oscillation to cause the second term on the right-hand side to be non-vanishing.

To do this, we observe that (10) provides an upper bound on ; indeed on any fixed compact set in , we have . This one-sided bound on the Laplacian is enough to rule out the oscillation problem. Indeed, as converges locally uniformly to , we see that

(29)

for any non-negative bump function and chosen so that on the support of . Integrating by parts and disposing of a lower order term, we conclude that

. (30)

On the other hand, since is bounded converges weakly to zero, one has

. (31)

One can easily replace and here by and . Combining (30) and (31) we conclude that the second term on the RHS of (28) is non-positive in the sense of distributions. But it is clearly also non-negative, and so it vanishes as required.

This gives (11); as a by-product of the argument we have also established the useful fact

(32)

in the sense of distributions. Combining this with the growth bounds (26), (27) on and $l_\infty$ from the previous section (which give exponential decay bounds on , and their first derivatives), it is not too difficult to then justify the remaining steps (12)-(18) of the argument rigorously; see Section 9.2 of Morgan-Tian for full details. (Note that once one reaches (13), one has a nonlinear heat equation for , and it is not difficult to use the smoothing effects of the heat kernel to then show that the locally Lipschitz function is in fact smooth.)

– The asymptotic gradient shrinking soliton is not flat –

Finally, we show that the asymptotic gradient shrinking soliton is non-trivial in the sense that its curvature is not identically zero at some time. For if the curvature did vanish everywhere at time t, then the equation (18) simplifies to . On the other hand, being flat, is the quotient of Euclidean space by some discrete subgroup. Lifting up to this space, we thus see that f is quadratic, and more precisely is equal to plus an affine-linear function. Thus f has no periodicity whatsoever and so the above-mentioned discrete subgroup is trivial. If we now apply (8) we see that . But on the other hand, as the original -solution was not flat, its reduced volume was strictly less than by Theorem 1 from Lecture 14, a contradiction. Thus the asymptotic gradient soliton is not flat.

– Appendix: Shi’s derivative estimates –

The purpose of this appendix is to prove the following estimate of Shi.

Theorem 3. Suppose that is a complete d-dimensional Ricci flow on the time interval , and that on the cylinder one has the pointwise curvature bound . Then on any slightly smaller cylinder one has the curvature bounds for any , where the implied constant depend on .

Proof. (Sketch) We induct on k. The case is trivial, so suppose that and that the claim has already been proven for all smaller values of k. We allow all implied constants in the O() notation to depend on . We refer to and as the “large cylinder” and “small cylinder” respectively.

We make some reductions. It is easy to see that we can take and T to be small.

Since on the cylinder, we see that the metric at later times of the large cylinder is comparable to the initial metric up to multiplicative constants. The curvature bound tells us that if is small, then we are inside the conjugacy radius; pulling back under the exponential map, we may thus assume that the exponential map from is injective on the large cylinder. Let be the time-varying radial coordinate; observe that the annulus will be contained between the large cylinder and small cylinder for T small enough.

Exercise 5. Show that if and T are small enough, then on the large cylinder.

Let be a smooth non-negative radial cutoff to the large cylinder that equals 1 on the small cylinder. From the above exercise, the Gauss lemma, and the chain rule, we see that .

Now we study the heat equation obeyed by the “energy densities” for various m.

Exercise 6. (Bochner-Weitzenböck type estimate) For any , show that

. (33)

(Hint: start with the equation and use the product rule and the definition of curvature repeatedly.)

From this exercise and the induction hypothesis we see that

(34)

for all , with the understanding that the second term on the right-hand side is absent when m=0. Telescoping this, we can thus find an expression

(35)

for some sufficiently large positive constant , which obeys the heat equation . Also, by hypothesis we have E=O(1) at time zero. Applying the maximum principle, we obtain the claim.

Exercise 6. Suppose that in the hypotheses of Shi’s theorem that we also have for on the large cylinder at time zero. Conclude that we have on the small cylinder for all j.

Exercise 7. Let be a smooth compact manifold, and let be a bounded solution to the heat equation which obeys a pointwise bound at time zero. Establish the bounds on the spacetime and all , where the implied constant depends on (M,g), K, T, and k.

(Update, June 2: some corrections.)

(Update, Oct 18 2011: A recently discovered issue with the Hamilton compactness theorem in the case where the curvature bound is not uniform in r has been addressed.)

The main idea here is to exploit what I have called the infinite convergence principle in a previous post: that every bounded monotone sequence converges. In the context of -solutions, we can apply this principle to either of our monotone quantities: the Perelman entropy

(1)

where is the backwards time variable and

, (2)

or the Perelman reduced volume

(3)

where is a fixed base point. As pointed out in Lecture 11, these quantities are related, and both are non-increasing in .

The reduced volume starts off at when , and so by the infinite convergence principle it approaches some asymptotic limit as . (We will later see that this limit is strictly between 0 and .) On the other hand, the reduced volume is invariant under the scaling

, (4)

in the sense that

. (5)

Thus, as we send , the reduced volumes of the rescaled flows (which are also -solutions) converge pointwise to a constant .

Suppose that we could somehow “take a limit” of the flows (or perhaps a subsequence of such flows) and obtain some limiting flow . Formally, such a flow would then have a constant reduced volume of . On the other hand, the reduced volume is monotone. If we could have a criterion as to when the reduced volume became stationary, we could thus classify all possible limiting flows , and thus obtain information about the asymptotic behaviour of -solutions (at least along a subsequence of scales going to infinity).

We will carry out this program more formally in the next lecture, in which we define the concept of an asymptotic gradient-shrinking soliton of a -solution.
In this lecture, we content ourselves with a key step in this program, namely to characterise when the Perelman entropy or Perelman reduced volume becomes stationary; this requires us to revisit the theory we have built up in the last few lectures. It turns out that, roughly speaking, this only happens when the solution is a gradient shrinking soliton, thus at any given time one has an equation of the form for some and . Our computations here will be somewhat formal in nature; we will make them more rigorous in the next lecture.

The material here is largely based on Morgan-Tian’s book and the first paper of Perelman. Closely related treatments also appear in the notes of Kleiner-Lott and the paper of Cao-Zhu.

– Stationarity of the Perelman entropy –

We begin with a discussion of the Perelman entropy, which is simpler than the Perelman reduced volume but which will serve as a model for the latter. To simplify the exposition we shall argue at a formal level, assuming all integrals converge, that all functions are smooth, all infima are actually attained, etc.

In Exercise 9, we already saw that if solves the nonlinear backwards heat equation

(6)

then the quantity obeyed the monotonicity formula

(7)

where H is the non-negative quantity

. (8)

In terms of the function , we also recall that (6) can be rewritten as the adjoint heat equation . In particular, we see that if is ever stationary at some time , then the solution must obey the gradient shrinking soliton equation

(9)

at that time . Using the uniqueness properties of Ricci flow (and of the backwards heat equation), one can then show that (9) persists for all subsequent times. Formally at least, this argument also shows that the Perelman reduced entropy can only be stationary on gradient shrinking solitons.

Let us analyse the monotonicity formula (7) further. If we write

(10)

then (7) asserts that

(11)

Since , we thus see that must equal plus a quantity which integrates to zero (i.e. a divergence). Given this, and given the fact that u (which is a close relative to v) obeys the adjoint heat equation), the following fact is then not so surprising:

Exercise 1. With the above assumptions, show that v obeys the forced adjoint heat equation

. (12)

– Stationarity in the Bishop-Gromov reduced volume –

Before we turn to the monotonicity of the Perelman reduced volume, we first consider the simpler model case of the Bishop-Gromov reduced volume (Corollary 1 of Lecture 9). An inspection of the proof of that result reveals that the key point was to establish the pointwise inequality

(13)

on a manifold of non-negative Ricci curvature , where for some fixed origin . To simplify the exposition let us assume we are inside the injectivity radius, and away from the origin, to avoid any issues with lack of smoothness.

We gave a proof of (13) in Lecture 10 using the second variation formula

(14)

whenever is a geodesic at , with and ; (see equation (17) of Lecture 10). From this (and the first variation formula) we obtain the inequality

(15)

for any vector field Y along the minimising geodesic from to x that equals 0 at and equals v at .

Of course, the only way that (15) can be an equality is if Y minimises the right-hand side subject to the constraints just mentioned. A standard calculus of variations computation lets one extract the Euler-Lagrange equation for this variational problem:

Exercise 2. Show that if (15) is obeyed with equality, then Y must obey the Jacobi equation

. (16)

Vector fields obeying (16) are known as Jacobi fields.

Recall from Lecture 10 that the inequality (13) was derived by applying (15) for v in an arbitrary orthonormal frame, and with , where v was extended by parallel transport along (thus ). Thus, in order for (13) to be obeyed with equality, the fields must be a Jacobi field for each v. Applying (16), and noting that , we conclude that we must have

(17)

along in order for (13) to be obeyed with equality. The converse is also true:

Exercise 3. Establish the identity

(18)

in the injectivity region, and conclude (13) is true with equality whenever (17) holds along the minimising geodesic .

As a consequence of the above analysis, we see that the Bishop-Gromov reduced volume can only be stationary on a sphere when (17) holds on the ball within that sphere.

We can also use the theory of Jacobi fields to get a more precise formula for (and hence ). The key observation is that the Jacobi equation (16) can be written as the linearisation

(19)

of the geodesic equation . This is ultimately unsurprising, since the geodesic equation and the Jacobi equation come from the Euler-Lagrange equations for the energy functional and a quantity related to a variation of the energy functional. But it allows us (at least inside the injectivity region, which also turns out (again, unsurprisingly) to be the region where the boundary value problem for the Jacobi equation always has unique solutions), to view Jacobi fields as the infinitesimal deformation field of geodesics.

Now let be a family of geodesics from to , so that and so (by (19)) Y is a Jacobi field for each s with Y(s,0)=0 and . (In general one no longer expects to have Y be geodesic in the s direction, i.e. need not be zero, but this will not concern us.) The first variation formula (i.e. the Gauss lemma ) then gives

(20)

and differentiating this again gives

(21)

Expanding out the left-hand side by the product rule and using (20) and the torsion-free identity we conclude the second variation formula

(22)

whenever is a Jacobi field along the minimal geodesic from to with Y(0)=0 and Y(1)=v, and whenever one is inside the injectivity region.

Exercise 4. Let Y be a Jacobi field with Y(0)=0 and Y(1)=v, and suppose one is inside the injectivity region. Use (22) and (16) to show that (15) in fact holds with equality, thus providing a converse to Exercise 2. (Hint: apply the fundamental theorem of calculus to the right-hand side of (22).)

– Constancy of the Perelman reduced volume –

We can obtain parabolic analogues of the above elliptic arguments to conclude when the Perelman reduced volume is stationary. Again, let us argue formally and assume that we are working inside the injectivity domain from a point .

Write . Recall from Lecture 10 that the proof of monotonicity of reduced volume relied on the inequality

(23)

which in turn followed from the three equalities and estimates

(24)

(25)

. (26)

Thus, in order for the reduced volume to be stationary at some time , one must have (23) (or equivalently, (26)) holding with equality throughout M at this time.

It is convenient to normalise . Recall from Lecture 10 that the proof of (26) proceeded via the second variation formula

(27)

applied to the vector field , where v obeys the ODE

. (28)

As in the elliptic case, equality in (26) can only hold if Y obeys the Euler-Lagrange equation for the right-hand side of (23), which can be computed to be

. (29)

Solutions of (29) are known as -Jacobi fields. As in the elliptic case, this equation can be rewritten as the linearisation

(30)

of the -geodesic equation , where

(31)

was introduced in Lecture 10.

Exercise 5. Verify (29) and (30).

If is now a smooth family of minimising -geodesics from to , then the variation field is an -Jacobi field by (30) (and conversely, inside the region of injectivity, any Jacobi field on a minimising geodesic can be extended locally to such a smooth family. The first variation formula (24) gives

(32)

where , and so on differentiating again and arguing as in the elliptic case we obtain

(33)

whenever Y is an -Jacobi field with Y(0)=0 and Y(1)=v.

Exercise 6. Show (using (31) and the fundamental theorem of calculus, as in Exercise 4) that (33) is equal to (27).

Now we return to our analysis of when the reduced volume is stationary at . We had found in that case that the vector field , where v solved (28), must be a Jacobi field. Combining this with (33) we conclude that

(34)

for any v, or in other words that

. (35)

This is for time ; rescaling the above analysis gives more generally that

. (36)

We thus conclude (formally, at least) that whenever the reduced volume is stationary, then the manifold is a gradient shrinking soliton (at that instant in time, at least) with potential function given by the reduced length. (The computation is only formal at present, because we have not addressed the issue of what to do on the -cut locus.)

Exercise 7. If (26) is obeyed with equality, show that the function obeys (6) and that (cf. the computations at the end of Lecture 11). From this and (7), deduce another (formal) proof of (36) whenever the reduced volume is stationary on an open time interval.

Remark 1. We have just seen that in the case of stationary reduced volume, the function f that appears in the entropy functional can be taken to be equal to the reduced length l. In general, one can take f to be a function bounded from above by the reduced length; see Corollary 9.5 of Perelman’s paper.

– Ricci flows of maximal reduced volume –

Recall that the reduced volume is equal to in the case of Euclidean space, and converges to this value in the limit in the case of complete Ricci flows of bounded curvature
(this can be shown by an analysis of the -exponential map for small values of , as discussed in Lecture 11). From this and the monotonicity of reduced volume we conclude that

(37)

for all such flows. We now characterise when equality occurs:

Theorem 1. Suppose that is a connected Ricci flow of bounded curvature on for some , such that (37) is obeyed with equality at the initial time for some point . Then M is Euclidean.

Proof. We give a sketch here only; full details can be found in Proposition 7.27 of Morgan-Tian’s book.

An inspection of the proof of monotonicity of reduced volume (especially as viewed through the -exponential map, as in Lecture 11) reveals that the domain of injectivity of the exponential map must have full measure, otherwise there will be a loss of reduced volume. The previous analysis then reveals that the equation (32) must hold outside of the cut locus; as l is Lipschitz and the manifold is smooth, one can then take limits and conclude that (32) holds globally (and so l is in fact smooth).

Combining (32) with the Ricci flow equation we obtain

, (38)

thus the metric is shrinking and also deforming by a vector field. In particular this gives an analogous equation for the magnitude of curvature (see equations (22), (26)):

. (39)

A maximum principle argument (which of course works in the absence of the dissipation term) then shows that if is strictly positive at one time, then it blows up as (like , in fact), which is absurd; and so this supremum must always be zero. In other words, the manifold is flat, and is therefore the quotient of by some discrete subgroup. But as the exponential map is almost always in the injectivity domain, this subgroup must be trivial, and the claim follows.

]]> http://terrytao.wordpress.com/2008/05/19/285g-lecture-13-li-yau-hamilton-harnack-inequalities-and-%ce%ba-solutions/ Mon, 19 May 2008 18:12:39 +0000 Terence Tao http://terrytao.wordpress.com/2008/05/19/285g-lecture-13-li-yau-hamilton-harnack-inequalities-and-%ce%ba-solutions/ We now turn to the theory of parabolic Harnack inequalities, which control the variation over space and time of solutions to the scalar heat equation

(1)

which are bounded and non-negative, and (more pertinently to our applications) of the curvature of Ricci flows

(2)

whose Riemann curvature or Ricci curvature is bounded and non-negative. For instance, the classical parabolic Harnack inequality of Moser asserts, among other things, that one has a bound of the form

(3)

whenever is a bounded non-negative solution to (1) on a complete static Riemannian manifold M of bounded curvature, are spacetime points with , and is a constant which is uniformly bounded for fixed when range over a compact set. (The even more classical elliptic Harnack inequality gives (1) in the steady state case, i.e. for bounded non-negative harmonic functions.) In terms of heat kernels, one can view (1) as an assertion that the heat kernel associated to dominates (up to multiplicative constants) the heat kernel at .

The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the -geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send ). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the -solutions introduced in the previous lecture.

In this current lecture, we shall discuss all of these inequalities (although we will not give the full details for the proof of Hamilton’s Harnack inequality, as the computations are quite involved), and derive several important consequences of that inequality for -solutions. The material here is based on several sources, including Evans’ PDE book, Müller’s book, Morgan-Tian’s book, the paper of Cao-Zhu, and of course the primary source papers mentioned in this article.

– Scalar parabolic Harnack inequalities –

Before we turn to the inequalities for Ricci flows (which are our main interest), we first consider the simpler case of scalar non-negative bounded solutions to the heat equation (1) on a static complete smooth Riemannian manifold . This case will not actually be used in our applications but serve as an important motivating example of the method. Our basic tools will be the scalar maximum principle and the following identity.

Exercise 1. Let be a smooth function. Establish the Bochner formula

. (4)

(Hint: use abstract index notation, and use the torsion-free nature of the connection, combined with the definitions of Riemann and Ricci curvature.)

This leads to the following consequence:

Exercise 2. Let be a strictly positive solution to (1), and let . Establish the nonlinear heat equation identities

(5)

(6)

. (7)

Now we can state the Li-Yau Harnack inequality.

Proposition 1. (Li-Yau Harnack inequality) Let M be a smooth compact d-dimensional Riemannian manifold with non-negative Ricci curvature, and let be a strictly positive smooth solution to (1). Then for every , we have

. (8)

Proof. By adding an epsilon to u if necessary (and then sending epsilon back to zero at the end of the argument) we may assume that for some . (We shall use this trick frequently in the sequel and refer to it as the epsilon-regularisation trick.) Write and . From Cauchy-Schwarz we have , and so from (6) we see that F is a supersolution to a nonlinear heat equation:

. (9)

On the other hand, is a sub-solution to the same equation, and the hypothesis that u is smooth and bounded below by (together with the compactness of M) implies that F dominates at times close to . Applying the scalar maximum principle (Corollary 1 from Lecture 3) we conclude that . The claim (8) now follows from (5) and the chain rule.

Remark 1. One can extend this inequality to the case when M is not compact, but is instead complete with bounded curvature, as long as one now adds the hypothesis that u is bounded (which was automatic in the compact case). The basic idea used to modify the proof is to multiply u by a suitable weight that grows at infinity to force the minimum value of F to lie in a compact set so that the maximum principle arguments can still be applied; we omit the standard details.

Remark 2. Observe that when is Euclidean and u is the fundamental solution for some , that (8) becomes an equality.

For strictly positive ancient solutions to (1) on a compact manifold of non-negative Ricci curvature, one can send to negative infinity, we conclude from (8) that

. (10)

In particular we see that ; thus non-negative ancient solutions to the linear heat equation on compact manifolds of non-negative Ricci curvature are non-decreasing in time. [Actually, it turns out that the only such solutions are in fact constant, but we will shortly generalise this assertion to less trivial situations.]

One can linearise the inequality (10) in u, obtaining the assertion that

(10′)

for any vector field X. Indeed (10) and (10′) are easily seen to be equivalent by the Cauchy-Schwarz inequality. One advantage of the formulation (10′) is that it also holds true when u is merely non-negative, as opposed to strictly positive u, by the epsilon-regularisation trick. In terms of , (10′) can also be expressed as

(11)

although now one needs u to be strictly positive for (11) to make sense.

Now let be points in spacetime with , let , and let be a path from to . From the fundamental theorem of calculus and the chain rule we have

(12)

where and the integrand is evaluated at . Applying (11) and then exponentiating we obtain the Harnack inequality

(13)

which can be extended from strictly positive solutions u to non-negative solutions u by the epsilon-regularisation trick. (Observe the similarity here with the -geodesic theory from Lecture 10.) By choosing to be the constant-speed minimising geodesic from to , we thus conclude that

. (14)

Remark 3. Specialising to the case when u is a static harmonic function and sending (and using Remark 1), we recover a variant of Liouville’s theorem: a bounded harmonic function on a Riemannian manifold of bounded non-negative curvature is constant.

Exercise 3. If the non-negative solution u to (1) is not ancient, but is only restricted to a time interval , show that one still has the variant

. (15)

Exercise 4. If the non-negative solution u to (1) is restricted to a time interval , and one no longer assumes that the Ricci curvature is non-negative (but it will still be bounded, since M is compact), establish the Harnack inequality (3) for some . (Hint: repeat the above arguments but with F replaced by for some small . Show (using (6), (7)) that if is small enough, then obeys an inequality similar to (9) but with an additional factor of on the right-hand side.

Exercise 5. Establish the strong maximum principle: if M is compact and is a non-negative solution to (1) which is not identically zero, then it is strictly positive for times (or equivalently, if u vanishes at even one point in , then it is identically zero).

Exercise 6. Generalise the strong maximum principle to the case when u is a supersolution to the heat equation rather than a solution. Also generalise it to the case when the metric g is not static, but instead varies smoothly in time. (For an additional challenge, generalise further to the case when M is complete, the metric has uniformly bounded Riemann curvature, u is bounded, and one also has a drift term on the right-hand side of the equation for some bounded X.)

Exercise 7. Using the final generalisation of Exercise 6, as well as the evolution equation for scalar curvature (equation (31) of Lecture 1), show that the scalar curvature of a solution is strictly positive at every point in spacetime. (We will prove stronger versions of this fact later in this lecture.)

Further variants and applications of these scalar Harnack inequalities can be found in the paper of Li and Yau.

– Parabolic Harnack inequalities for the Ricci flow –

Now we turn from the scalar equation (1) to the Ricci flow equation (2), which one could think of as a kind of tensor-valued quasilinear heat equation (by de Turck’s trick, see Lecture 1). To begin with let us first consider the simple two-dimensional case d=2. In this case the Bianchi identities make the Riemann, Ricci, and scalar curvatures are all essentially equivalent (see Lecture 0); in particular one has the identity

(16)

in the two-dimensional case. In particular, the heat equation for scalar curvature (equation (31) from Lecture 1) simplifies to

(17)

in this case; compare this with (1).

Suppose that the scalar curvature R is strictly positive. Setting , one has an analogue of (5):

. (18)

Exercise 8. If we set , show the following analogue of (9):

. (19)

[Hint: you will need to first derive the identity for arbitrary smooth v.] Conclude that if M is compact and R is strictly positive on the time interval , then , and thus conclude Hamilton’s Harnack inequality for surfaces:

. (20)

Extend this inequality to the case when R is merely non-negative rather than strictly positive by setting f equal to rather than and then setting to zero (this is how should performs the epsilon regularisation trick for Ricci flow, by modifying the logarithm function by an epsilon, rather than the solution).

For ancient two-dimensional solutions with non-negative curvature, we thus conclude from the Harnack inequality (20) that R (and ) obeys the same bounds (10), (10′), (11) that scalar solutions u did previously. In particular R is non-decreasing in time, and more generally

(21)

for any X. We can also obtain an analogue of (13). Also, observe from the assumption of non-negative curvature and (2) that the metric is non-increasing with time, and so we can also deduce an analogue of (14):

. (22)

With a non-trivial amount of effort, one can extend Hamilton’s Harnack inequality to higher dimensions. One cannot argue solely using the scalar curvature R, because the equation for that curvature also involves the Ricci tensor, which thus also needs to be controlled. What is worse, one cannot argue solely using the Ricci tensor either, because the equation

(23)

for the evolution of that curvature involves the Riemann tensor. To proceed, one in fact has to deal with the equation for the full Riemann tensor,

(24)

where is an explicit but rather complicated quadratic expression in the Riemann curvature. This expression simplifies when using a moving orthonormal frame, as was done in Lecture 3, to the form

. (25)

By using (25) and many tensor calculations, one can (eventually) establish a (rather complicated) analogue of the (21) for , and hence for and then (after taking some traces) to R. In particular, we have

Theorem 1. (Hamilton’s Harnack inequality for ancient Ricci flows) Let be a complete ancient Ricci flow with non-negative bounded Riemann curvature. (In particular, all -solutions are of this form.) Then we have the pointwise inequality

(26)

for any vector field X.

Note that in the two-dimensional case, (26) collapses to (23) thanks to (16).

The proof of (26) is remarkably delicate (in particular, going through the tensor curvature equation (25)), but ultimately follows broadly similar lines to the previous arguments (i.e. Bochner-type identities, Cauchy-Schwarz type inequalities, and tensor maximum principles). For technical reasons it is also convenient to carry auxiliary tensor fields such as the vector field X appearing in (26) throughout the argument. We refer the reader to Hamilton’s original paper for details. (There are alternate proofs, such as the one by Chow and Chu using a metric closely related to the high-dimensional metrics considered in Lecture 9, but all of the proofs I know of require a significant amount of calculation.)

Exercise 9. Suppose that solves the gradient steady soliton equation for some smooth f. Using the Bianchi identity , establish the identity

(27)

(note this identity also holds for gradient shrinking or expanding solitons) and then by taking divergences and using the Bianchi identity again, establish that

. (28)

Conclude that (26) is an identity in this case when one sets .

– Applications of the Harnack inequality –

Now we develop some applications of the Harnack inequality for -solutions. One easy application follows by setting X equal to 0, giving
the pointwise monotonicity of the scalar curvature in time:

. (29)

Another application is to obtain a slightly weakened version of (22) (with the 4 in the denominator replaced by 2):

Exercise 10. Show that one has whenever one has non-negative Riemann curvature. Using this and (26), show that

. (30)

for all -solutions and all spacetime points with .

Now we use the Harnack inequality to obtain some further control on the reduced length function . Recall that this quantity takes the form

(31)

where and is a minimising -geodesic, which in particular means that it obeys the -geodesic equation

(32)

(see equation (27) from Lecture 10). Using (32) and the chain rule, we can compute the total derivative along the path
as

. (33)

On the other hand, the Harnack inequality (26) (with X replaced by 2X) lets us bound the total derivative of R:

. (34)

We add (33) and (34) and rearrange to obtain

. (35)

We (somewhat crudely) discard the non-negative term and integrate in using (31) to obtain

(36)

where we abbreviate as l. Using the first variation formulae for reduced length (see equations (14), (15) from Lecture 11), as well as the nonnegativity of R (and hence of l), we obtain the useful inequalities

(37)

and

. (38)

Informally, this means that at any given point to the past of , l is roughly constant at spatial scales and at temporal scales . Furthermore, if l is bounded, then one has bounded normalised curvature at such scales.

– A splitting theorem –

Our final application of these ideas (or more precisely, of the strong maximum principle) will be to establish a dichotomy (due to Hamilton) for 3-dimensional -solutions: either their Ricci curvature is strictly positive, or the solution splits locally as the product of a line with a two-dimensional solution.

Proposition 1. Let be a three-dimensional -solution. Suppose that the Ricci tensor has a zero eigenvalue at some point . Then on the slab , the Ricci flow locally splits as the product of a two-dimensional Ricci flow and a line.

Proof. The first stage is to show that the Ricci tensor has a zero eigenvalue on all of
. Let denote the three eigenvalues of the Riemann tensor as viewed in an orthonormal frame (as in Lecture 3), thus a zero eigenvalue of the Ricci tensor is equivalent to . Suppose for contradiction that at some time , this quantity is not identically zero, thus we can find some non-negative scalar function , not
identically zero, such that at time . We then extend h by the heat equation, so by the strong maximum principle h is strictly positive for all times after . From the convexity of the functional (which one can view as the minimal trace of over two-dimensional subspaces), we see that the set cuts out a fibrewise convex parallel subset of a suitable vector bundle over (in the sense of the tensor maximum principle, Proposition 1 from Lecture 3), which one can easily check to be preserved under the ODE associated to the simultaneous evolution of (25) and the scalar heat equation for h.
Applying the tensor maximum principle we conclude that for all times in , and in particular that is non-zero at , a contradiction. Thus the Ricci curvature must have a zero eigenvalue on all of , thus on this slab. On the other hand, from Exercise 7 we must have throughout this slab.

The symmetric rank 2 tensor thus has rank 1 at every point, and thus locally can be expressed in the form for some smooth non-zero scalar a and a unit vector field v. (If M was orientable, one could extend this vector field to be global). The equation (25) then becomes

. (39)

Since v is a unit vector field, the vector fields are orthogonal to v for every v. Thus we can restrict to the component of (39) that is completely orthogonal to v, and conclude (since a is nonzero) that . If we then inspect the component of (39) which is partially orthogonal to v, we also learn that . Expressing the left-hand side in an orthonormal basis as the sum of rank one positive semi-definite matrices, we easily conclude that , i.e. v is parallel to the connection. This implies that the dual one-form is closed and hence locally exact; thus v is locally the gradient of some potential function f. From this we easily see that the flow locally splits as the product of a two-dimensional flow (on a level set of f) and a line (the flow lines of v), and then it is easy to verify that the two-dimensional flow is a Ricci flow, as claimed.

Remark 4. One cannot always extend this local splitting to a global one, due to topological obstructions; consider for instance the oriented round shrinking cylinder quotient (Example 3 of Lecture 12). One could also imagine the product of a round shrinking and a static circle , in which the null eigenvector of the Ricci tensor splits off as a circle rather than a line; but this is not a -solution because it becomes collapsed at large scales in the distant past.

Remark 5. The above splitting analysis can be carried out in any dimension; for instance, one can show that the rank of the Riemann tensor is a constant for any ancient solution with bounded non-negative Riemann curvature. For this and further splitting results in this case, see the paper of Hamilton.

-solutions

[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]

The next few lectures will be devoted to the analysis of -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about -solutions.

Our treatment here is based primarily on the book of Morgan and Tian.

– Definition of a -solution –

We fix a small number (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a -solution:

Definition 1. (-solutions) A -solution is a Ricci flow which is

1. Ancient, in the sense that t ranges on the interval ;
2. Complete and connected (i.e. (M,g(t)) is complete and connected for every t);
3. Non-negative Riemann curvature, i.e. is positive semidefinite at all points in spacetime;
4. Bounded curvature, thus ;
5. -noncollapsed (see Definition 1 of Lecture 7) at every point in spacetime and at every scale ;
6. Non-flat, i.e. the curvature is non-zero at at least one point in spacetime.

This laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.

Remark 1. If a d-dimensional Riemann manifold is both flat (thus ) and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space . Thus the non-flat condition is only excluding the trivial Ricci flow with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere.

Remark 2. In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and are non-negative, and that and . Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of -solutions are controlled by the scalar curvature.

Now we discuss examples (and non-examples) of -solutions.

Example 1. Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a -solution for sufficiently small if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere with the standard metric is a gradient shrinking solution and will generate a -solution for any and sufficiently small , which we will refer to as the shrinking round sphere -solution.

Exercse 1. Show that the Cartesian product of two -solutions is again a -solution (with a smaller value of ), as is the Cartesian product of a -solution. Thus for instance the product of the shrinking round 2-sphere and the Euclidean line is a -solution, which we refer to as the shrinking round 3-cylinder .

Example 2. In one dimension, there are no -solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is not a -solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere is -solution, as discussed above. We can quotient this by the obvious action to also get a shrinking round projective plane as a -solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional -solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder is not a -solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a -solution due to it being collapsed at large scales.

Example 3. In three dimensions, we begin to get significantly more variety amongst -solutions. We have the round shrinking 3-sphere , but also all the quotients of such round spheres by free finite group actions (including the projective space , but with many other examples. We refer to these examples as round shrinking 3-spherical space forms. We have also seen the shrinking round cylinder ; there are also finite quotients of this example such as shrinking round projective cylinder , or the quotient of the cylinder by the orientation-preserving free involution . We refer to these examples as the unoriented and oriented quotients of the shrinking round 3-cylinder respectively. The oriented quotient can be viewed as a half-cylinder capped off with a punctured (and the whole manifold is in fact homeomorphic to a punctured ).

Example 4. One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking or shrinking example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as C-components (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders capped off with what is topologically either a punctured or punctured (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as -capped strong -tubes (we will define this term precisely later). One can also consider doubly -capped strong -tubes, in which an approximately round finite cylinder is capped off at both ends by either a punctured or punctured ; such manifolds then become homeomorphic to either or . (Note we need to cap off any ends that show up in order to keep the manifold M complete.)

An important theorem of Perelman shows that these examples of -solutions are in fact the only ones:

Theorem 1. (Perelman classification theorem, imprecise version) Every 3-dimensional -solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):

1. A shrinking round 3-sphere (or shrinking round spherical space form );
2. A shrinking round 3-cylinder , the quotient , or one of its quotients (either oriented or unoriented);
3. A C-component;
4. A C-capped strong -tube;
5. A doubly C-capped strong -tube.

We will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).

Remark 3. At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong -tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong -tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional -solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as , or as the product of the cigar soliton and a line) or 3-dimensional (as in ) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not -solutions. The unoriented quotiented cylinder also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur.

Remark 4. In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round with the trivial plane . This is a -solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).

The classification theorem lets one understand the geometry of neighbourhoods of any given point in a -solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):

Definition 2. (Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.

1. We say that U is an -neck if it is close (in a smooth topology) to a round cylinder , with x well in the middle of of this cylinder;
2. We say that U is a C-component if U is diffeomorphic to or (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.
3. We say that U is -round if it is close (in a smooth topology) to a round sphere or spherical space form (i.e. it is close to a constant curvature manifold).
4. We say that U is a -cap if it consists of an -neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)
5. We say that U is a canonical neighbourhood of x if it is one of the above four types.

When the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.

Using Theorem 1 (and defining all terms precisely), one can easily show the following important statement:

Corollary 1 (Canonical neighbourhood theorem for -solitons, informal version) Every point in a 3-dimensional -solution that does not contain an embedded copy of with trivial normal bundle is contained in a canonical neighbourhood.

The next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.

– High curvature regions of Ricci flows –

Corollary 1 is an assertion about -solutions only, but it implies an important property about more general Ricci flows:

Theorem 2. (Canonical neighbourhood for Ricci flows, informal version) Let be a Ricci flow of compact 3-manifolds on a time interval , without any embedded copy of with trivial normal bundle. Then every point with sufficiently large scalar curvature is contained in a canonical neighbourhood.

(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)

The importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)

Theorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:

1. Suppose for contradiction that Theorem 2 failed. Then one could find a sequence of points with which were not contained in canonical neighbourhoods.
2. M, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.
3. On any compact time interval , the scalar curvature is necessarily bounded, and thus . As a consequence, if we define the rescaled Ricci flows , where is the natural length scale associated to the scalar curvature at , then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds may cease to be compact, but will remain complete.
4. Because of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3), we expect the rescaled flows to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).
5. If we can pick the points suitably (so that the scalar curvature is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows have bounded curvature in the limit.
6. Since -noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7) should ensure that the rescaled flows remain -noncollapsed in the limit.
7. Since the rescaled scalar curvature at the base point of is equal to 1 by construction, any limiting flow will be non-flat.
8. Various compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow . This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.
9. Applying 2-8, we see that the limiting flow is a -solution.
10. Applying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points also lie in canonical neighbourhoods for sufficiently large n, a contradiction.

There are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for -solutions which we will deduce in later lectures. For Step 5, the problem is that the points that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point (much further away than the natural length scale , in particular), so that it does not show up in the limiting flow . We will discuss these issues in more detail in later lectures.

– Benchmarks in controlling -solutions –

As mentioned earlier, the next few lectures will be focused on controlling -solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.

Proposition 1. (Consequences of Hamilton’s Harnack inequality) Let be a -solution. Then is a non-decreasing function of time. Furthermore, for any , we have the pointwise inequalities

(1)

and

(2)

on , where of course is the backwards time variable.

These inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of -solutions, as well as the Ricci flow equation of course.

Now one can handle the two-dimensional case:

Proposition 2. (Classification of 2-dimensional -solutions) The only two-dimensional -solutions are the round shrinking 2-spheres.

This proposition relies on first studying a certain asymptotic limit of the -solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original -solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.

Turning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.

Proposition 3. (Asymptotic curvature) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .

The proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.

The increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:

Proposition 4. (Asymptotic volume collapse) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .

Note that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.

An important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball at time zero on which the volume is large (e.g. larger than for some ), one has bounded normalised curvature, thus on this ball. This fact helps us deduce

Theorem 3. (Perelman compactness theorem, informal version) The space of all pointed -solutions (allowing to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.

One corollary of this compactness is that there is in fact a universal such that every -solution is a -solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on -solutions which come from the previous volume analysis.

The proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.

The route to -noncollapsing via reduced volume proceeds by the following scheme:

Non-collapsing at time t=0 (1)

Large reduced volume at time t=0 (2)

Large reduced volume at later times t (3)

Non-collapsing at later times t (4)

The implication is the monotonicity of Perelman reduced volume. In this lecture we discuss the other two implications , and ).

Our arguments here are based on Perelman’s first paper, Kleiner-Lott’s notes, and Morgan-Tian’s book, though the material in the Morgan-Tian book differs in some key respects from the other two texts. A closely related presentation of these topics also appears in the paper of Cao-Zhu.

– Definitions –

Let us first recall our definitions. Previously we defined Perelman reduced length and reduced volume for ancient flows for , centred at a point on the final time slice , but one can also define these quantities for flows on the time interval and for points as follows. We introduce the backward time variable . Given any path , we define its length

(5)

and for any with , with , we define the reduced length

(6)

where ranges over all paths from to (which can also be viewed as trajectories in the spacetime manifold from to . The reduced volume is then defined as

. (7)

[Note: some authors normalise the reduced volume by using instead of , in order to give Euclidean space a reduced volume of 1, but this makes no essential difference to the analysis.]

The arguments of the previous lecture show that if is a Ricci flow, then the reduced volume is a non-increasing function of for fixed . In particular, the reduced volume at later times is bounded from below by the reduced volume at time 0 (which is the implication ).

– Heuristic analysis –

In the case of the trivial Euclidean flow, the reduced length is given by the formula

(8)

with the minimising geodesic given by the formula

with (9)

Here, we briefly argue why we expect heuristically to have a similar relationship

(10)

for the reduced length on more general Ricci flows, under an assumption of bounded normalised curvature.

Specifically, suppose that we have a normalised curvature bound . Then we have , and so over the time scale , we see that the metric only changes by a multiplicative constant. If we ignore such constants for now, we see that the distance function does not change much over the time interval of interest.

Let be a minimising -geodesic from to . This path has to traverse a distance roughly in time , and so its speed should be at least . Also, the scalar curvature R should be by the bounded normalised curvature assumption. Putting all this into (5) and (6) we heuristically obtain (10).

From (10), we expect the expression to be comparable to 1 when is inside the ball , and to be exponentially small outside of this ball. Using (7), we thus obtain a heuristic approximation for the Perelman reduced volume:

. (11)

Thus the Perelman reduced volume is heuristically equivalent to the Bishop-Gromov reduced volume at at scale . Since the latter measures non-collapsing, we heuristically obtain the implications and .

– From non-collapsing to lower bounds on reduced volume –

Now we discuss implications of the form in more detail. Specifically, we show

Proposition 1. Let be a d-dimensional Ricci flow on a complete manifold M for such that we have the normalised initial conditions and at time t=0 for some and all x (so in particular, the geometry is non-collapsed at scale 1 at all points at time zero). Then we have for some and all .

The main task in proving implications of the form is to show the existence of some large ball at time zero on which is bounded from above.

Turning to the specific proposition above, we first observe that we can reduce to the large time case . Indeed, if , then we can rescale the Ricci flow until (this increases T, but we can simply truncate T to compensate for this). This rescaling reduces the size of the initial Riemann curvature, and the volume of balls of unit radius are still bounded from below thanks to the Bishop-Gromov inequality.

The next observation we need is that the control on the geometry at time zero persists for a short amount of additional time:

Lemma 1. (Local persistence of controlled geometry) Let the hypotheses be as in Proposition 1. Then there exists an absolute constant c > 0 (depending only on d) such that for all times and . Also we have for all and , and some depending only on .

Proof. We recall the nonlinear heat equation

(12)

for the Riemann curvature tensor under Ricci flow (see equation (31) of Lecture 1). The bound on Riemann curvature can then obtained by an application of Hamilton’s maximum principle (Proposition 1 from Lecture 3); we leave this as an exercise to the reader. [Technically, one needs to first generalise the maximum principle from compact manifolds to complete manifolds of bounded curvature. This can be done using barrier functions, but it is somewhat technically involved: see Chapter 12 of Chow et al. for details.] As in the heuristic discussion, the bounds on the Riemann curvature (and hence the Ricci curvature) show that the metric g and the distance function only change by at most a multiplicative constant; this also implies that the volume measure only changes by a multiplicative constant as well. From this we see that the lower bound on the volume of unit balls at time zero implies a lower bound on the volume of balls of radius O(1) at times ; one can then get back to balls of radius 1 by invoking the Bishop-Gromov inequality.

The next task is to find a point such that the reduced length from to is small, since this should force y (and the points close to y) to give a large contribution to the reduced volume. In the Euclidean case, one would just take (see (8)), but this does not necessarily work for general Ricci flows: note from (5), (6) that the reduced length from to could in principle be as large as

, (13)

which could be quite large if the scalar curvature becomes large and positive (which is certainly within the realm of possibility, especially if one is approaching a singularity).

Fortunately, we can use the parabolic properties of the reduced length , combined with the maximum principle, to locate a good point y with the required properties. From the analysis of the previous lecture, and some rescaling and time translation, we obtain the identities and inequalities

(14)

(15)

(16)

(cf. equations (29), (33), (47) from the previous lecture), where is the final velocity vector of the minimising -geodesic from to . [We only derived (14)-(16) rigorously inside the domain of injectivity, but as discussed in the previous lecture, one can establish the above inequalities in the sense of distributions on the whole manifold M.] From (15), (16) we obtain in particular that l is a supersolution of a heat equation:

. (17)

[Note that (17) holds with equality in the Euclidean case (8).] From the maximum principle (Corollary 1 from Lecture 3), we see that if we have the uniform lower bound at some time , then this bound will persist for all times between t and . On the other hand, by using the upper bound (12) for we see that the bound breaks down for times t sufficiently close to . We therefore conclude that for all . In particular we can find a point y such that

, (18)

where c is the small constant in Lemma 1. Given the bounded geometry control in Lemma 1 (and in particular the fact that g(t) is comparable to g(0) for ), it is thus not hard to see (by concatenating the minimising path from to with a geodesic segment (in the g(0) metric) from to ) that

(19)

for some depending only on d, where. The hypotheses on the geometry of g(0), combined with the Bishop-Gromov inequality, give a uniform lower bound for the volume of , and Proposition 1 now follows directly from the definition (7) of reduced volume.

– From lower bounds on reduced volume to non-collapsing –

Now we consider the reverse type of implication from those just discussed. Here, the task is reversed; rather than establishing upper bounds on l on a ball of radius comparable to one, the main challenge is now to establish lower bounds (of the form ) on l on such a ball, as well as some growth bounds on l away from this ball.

We begin by formally stating the result of the form that we shall establish.

Proposition 2. Let be a d-dimensional Ricci flow on a complete manifold M for , and let and be such that for and , and such that for some and all . Then one has for some c depending only on d and .

Exercise 1. Use Proposition 1, Proposition 2, and the monotonicity of Perelman reduced volume to deduce Theorem 2 from Lecture 7.

We now prove Proposition 2. We first observe by time translation (and by removing the portion of the Ricci flow below that we may normalise , and then by scaling we may normalise . Thus we now have a Ricci flow on [0,1] with on and

(20)

for all , where is the reduced length function. Our task is to show that is bounded away from zero.

We first observe (as in Lemma 1) that the metrics g(t) for are all comparable to each other up to multiplicative constants on , and so the balls in these metrics also differ only up to multiplicative constants.

Next, we would like to localise the reduced volume (20) to the ball (since this is the only place where we really control the geometry). To do this it is convenient to work in the parabolic counterpart of normal coordinates around and exploit the pointwise version of the Perelman reduced volume monotonicity. To motivate this, recall from the pointwise inequality

(21)

that we had the Bishop-Gromov inequality

(21′)

where is the sphere of radius r centred at with area element dS. Indeed, we can rewrite the left-hand side of (21′) as

(22)

where is the standard sphere with the standard area element , and is the Jacobian of the exponential map ; in the Euclidean case, . [Actually, once the radius r exceeds the injectivity radius, one has to restrict to the portion of that has not yet encountered the cut locus, but let us ignore this technical issue for now.] The inequality (21) (when combined with the Gauss lemma) is equivalent to the pointwise inequality

(23)

which certainly implies (22), but also implies the stronger fact that the Bishop-Gromov inequality can be localised to arbitrary sectors in the sense that (which can be viewed as the Bishop-Gromov reduced volume of the sector ) is non-increasing in .

Now we develop parabolic analogues of the above observations. Recall from the previous lecture that we have an -exponential map for that sends a tangent vector v to , where is the unique -geodesic starting at with initial condition . In the Euclidean case, this map is given by the formula

(24)

as can be seen from (9). We can then rewrite the reduced volume in terms of “normal coordinates” as

(25)

where is the Jacobian of the map . (Again, one has to restrict to the portion of the tangent manifold lies inside the injectivity domain, but this domain turns out to be non-increasing in (for much the same reason that the region inside the cut locus of a point in a Riemannian manifold is star-shaped) and so this effect works in our favour as far as monotonicity is concerned.)

In the previous lecture we saw that the monotonicity of Perelman reduced volume followed from the pointwise inequality

(26)

which of course also follows from (14)-(16).

Exercise 2. Use (14), (26), and the identity

(27)

(which basically follows from the fact that any segment of a minimising -geodesic is again a -geodesic) to derive the pointwise inequality

. (38)

Exercise 2 reproves the monotonicity of Perelman reduced volume (25), but also proves a stronger local version of this monotonicity in which the region of integration is replaced by an arbitrary region (intersected with the injectivity region, as mentioned earlier).

In the Euclidean case, a computation using (8) and (24) shows that and . Also, one can use some basic analysis arguments to show that in the limit , the expressions in (25) converge pointwise to their Euclidean counterparts. As a consequence we obtain the pointwise domination

(39)

for any v and any . As a consequence, the far part of (25) (corresponding to “fast” geodesics) is negligible: we have

(40)

for some C depending only on d and . From this and the hypothesis (19) we thus obtain lower bounds on local Perelman reduced volume, or more precisely that

(41)

for all .

Now, we have bounded curvature on the cylinder . Using the heat equation (12) and standard parabolic regularity estimates, we thus conclude that any first derivatives of the curvature are also bounded on the cylinder . (In fact, all higher derivatives are controlled as well; see this paper of Shi for full details.) In particular we have in this cylinder. Thus the equation G=0 for an -geodesic (where G was defined in equation (27) of the previous lecture) becomes

(42)

or equivalently that

(43)

as long as the geodesic stays inside this smaller cylinder. From this and Gronwall’s inequality one easily verifies that for sufficiently small (depending on C, d), the exponential map does not exit the cylinder for . On the other hand, at time , we see from (5), (6) and the bounds on curvature in this cylinder that the reduced length l of the associated -geodesic is bounded below by some constant depending on . We thus see (from the change of variables formula) that the left-hand side of (41) is bounded above by . Choosing to be a small number depending on C, d, we thus conclude from (41) that the volume of with respect to (and hence g(1), by comparability of metrics) is bounded from below by some constant depending on C and d, and thus ultimately on and d, giving Proposition 2 as desired.

– Extensions –

The pointwise nature of the monotonicity of Perelman reduced volume allows one to derive local versions of the non-collapsing result, in which one only needs a portion of the geometry to be non-collapsed at the initial time. A typical version of such a local noncollapsing result reads as follows.

Theorem 1 (Perelman’s non-collapsing theorem, second version) Let be a d-dimensional Ricci flow on the time interval , and suppose that one has the bounded normalised curvature condition on a cylinder for some . Suppose also that we have the volume lower bound for some . Then for any , the Ricci flow is -noncollapsed at for any and at any scale , for some depending only on d, c, A.

The novelty here is that the geometry is controlled in a cylinder, rather than on the initial time slice, but one gets to conclude -noncollapsing at points some distance away from the cylinder. In view of Lemma 1, we see that this result is more or less a strengthening of the previous -noncollapsing theorem.

This theorem (or more precisely, a generalisation of it involving Ricci flow with surgery) is used in the original argument of Perelman (and then in the later treatments by Kleiner-Lott and Cao-Zhu) in order to deal with the long-time behaviour of Ricci flow with surgery, which is needed for the geometrisation conjecture. For proving the Poincaré conjecture, though, one has finite time extinction, and it turns out that the above theorem is not needed for the proof of that conjecture (for instance, it does not appear in treatment of Morgan-Tian). Nevertheless I will sketch how the above theorem is proven below, since there are one or two interesting technical tricks that get used in the argument.

The proof of Theorem 1 is, unsurprisingly, a modification of the previous arguments . The implications and are basically unchanged, but one needs to replace Proposition 1 by the following variant.

Proposition 3. Let the hypotheses be as in Theorem 1. Then for any one has for some depending on A, c, d.

We sketch the proof of Proposition 3. It is convenient to rescale so that . In view of the non-collapsed nature of the geometry in , it suffices to establish a lower bound of the form for all for some depending on A,c,d. Actually, because of the bounded geometry in the cylinder, it suffices to show that for just one point for some depending on A,c,d, since one can join (1/2,y) by a geodesic to (1,z) much as in the proof of Proposition 1.

The task is now analogous to that of finding a point y that obeyed the relation (18), so we expect the heat equation (17) to again play a role. We do not need the sharp bound of n/2 which occurs in (18); on the other hand, y is now constrained to lie in a ball, which defeats a direct application of the maximum principle. To fix this one has to multiply the reduced length l by a penalising weight to force the minimum to lie in the desired ball at time 1/2, and then rapidly relax this weight as one moves from time 1/2 to time 1 so that it incorporates the point x at time 1. It turns out the maximum principle can then be applied with a suitable choice of weights, as long as one knows that the distance function is a supersolution to a heat equation, and more precisely that when r is bounded away from the origin. But this can be established by the first and second variation formulae for the distance function, and in particular using the non-negativity of the second variation for minimising geodesics. Details can be found in Section 8 of Perelman’s paper, Sections 26-27 of Kleiner-Lott, or Section 3.4 of Cao-Zhu.

Remark 1. One can also interpret the above analysis in terms of heat kernels, and using (26) instead of (17). The former inequality is equivalent to the assertion that the function is a subsolution of the adjoint heat equation: . As , v approaches a Dirac mass at x (indeed, v asymptotically resembles the Euclidean backwards heat kernel from ) and the task is to obtain upper bounds on v at some point on a ball at time 1/2. This is basically equivalent to establishing lower bounds of Gaussian type for the fundamental solution of the adjoint heat equation at some point in . Similar analysis in the case of a static manifold with potential (and a lower bound on Ricci curvature) was carried out somewhat earlier by Li and Yau.

As mentioned previously, in order to apply the non-collapsing result beyond the first surgery time, it is necessary to develop analogues of the above theory for Ricci flows with surgery. This turns out to be remarkably technical, but the main ideas at least are fairly clear. Firstly, one has to delete all -geodesics which pass through surgery regions when defining the Perelman reduced volume; such curves are called “inadmissible”. Note that if is in a surgery region to begin with, then every curve is inadmissible but in this case the geometry can be controlled directly from the surgery theory. As it turns out, one can similarly deal with the case when has extremely high curvature because one can control the geometry of such regions. So we can easily eliminate these bad cases.

Because of the pointwise nature of the monotonicity formula for reduced volume, this restriction of admissibility does not affect the “” stage of the argument. The “” step is also largely unaffected, since removing inadmissible components of the reduced volume only serves to strengthen the hypothesis (3). But significant new technical difficulties arise in the “” portion of the argument, when one has to argue that not too much of the reduced volume has been deleted by all the various surgeries that take place between time t=0 and time . In particular, we still need to find a point y obeying (18) (or something very much like (18)) which is admissible. To do this, the basic idea is to establish that inadmissible curves have large reduced length (and so removing them will not impact the search for a solution to (18)). For technical reasons it is better to restrict attention to barely admissible curves – curves which just touch the border of the surgery region, but do not actually enter it. In this case it is possible to use the geometric control of the surgery regions to give some non-trivial lower bounds on the reduced length of such curves, although there are still significant technical issues to resolve beyond this. I hope to return to this point later in the course, when we have defined surgery properly.

– Epilogue: a connection between Perelman entropy and Perelman reduced volume –

We have shown two routes towards establishing -non-collapsing of Ricci flows, one using the (parameterised) Perelman entropies

(44)

and one using the reduced volumes mentioned above. Actually, the two quantities are related to each other (this is hinted at in Section 9 of Perelman’s paper); very roughly speaking, the potential function f in the theory of Perelman entropy plays the same role that reduced length l does in the theory of Perelman volume. Indeed, using (44) and shifting f by a constant if necessary, we have the log-Sobolev inequality

. (45)

An integration by parts reveals that we can replace the on the left -hand side by , and hence one can also replace this quantity by .

We now apply this inequality with and for some spacetime point in the Ricci flow. Using (14), (16) we see that

(46)

and thus the left-hand side of (45) is non-positive. Using (7) we thus conclude a simple relationship between entropy and reduced volume:

. (47)

[As usual, we have equality in physical space; this inequality also reinforces the suggestion that one normalise the reduced volume by an additional factor of .]

Thus the Perelman entropy can be viewed as a global analogue of the Perelman reduced volume, in which we allow the base point to vary (thus it measures the global non-collapsing nature of the manifold, as opposed to the local nature; we already saw this in Lecture 8; compare in particular equation (62) from Lecture 8 with the heuristic (11) using (47).)

There are other connections between entropy and reduced volume; compare for instance the flow equation for the potential f (equation (46) from Lecture 8) with equation (26) here. The adjoint heat equation also makes essentially the same appearance in both theories. See Section 9 of Perelman’s paper for further discussion.

Remark 2. As remarked above, the flow equation for f can be viewed as a pointwise versions of the entropy monotonicity formula, which in principle leads to localised monotonicity formulae for the Perelman entropy; some analysis in this direction appears in Section 9 of Perelman’s paper. But I do not know if these localised entropy formulae can substitute to give a different proof of Theorem 1.

]]> http://terrytao.wordpress.com/2008/05/09/285g-lecture-10-variation-of-l-geodesics-and-monotonicity-of-perelman-reduced-volume/ Fri, 09 May 2008 15:40:23 +0000 Terence Tao http://terrytao.wordpress.com/2008/05/09/285g-lecture-10-variation-of-l-geodesics-and-monotonicity-of-perelman-reduced-volume/ Having completed a heuristic derivation of the monotonicity of Perelman reduced volume (Conjecture 1 from the previous lecture), we now turn to a rigorous proof. Whereas in the previous lecture we derived this monotonicity by converting a parabolic spacetime to a high-dimensional Riemannian manifold, and then formally applying tools such as the Bishop-Gromov inequality to that setting, our approach here shall take the opposite tack, finding parabolic analogues of the proof of the elliptic Bishop-Gromov inequality, in particular obtaining analogues of the classical first and second variation formulae for geodesics, in which the notion of length is replaced by the notion of -length introduced in the previous lecture.

The material here is primarily based on Perelman’s first paper and Müller’s book, but detailed treatments also appear in the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.

– Reduction to a pointwise inequality –

Recall that the Bishop-Gromov inequality (Corollary 1 from the previous lecture) states (among other things) that if a d-dimensional complete Riemannian manifold (M,g) is Ricci-flat (or more generally, has non-negative Ricci curvature), and is any point in M, then the Bishop-Gromov reduced volume is a non-increasing function of r. In fact one can obtain the slightly sharper result that is a non-increasing function of r, where is the sphere of radius r centred at .

From the basic formula (equation (1) from the previous lecture) and the Gauss lemma, one readily obtains the identity

(1)

where is the area element. The monotonicity of then follows (formally, at least) from the pointwise inequality

(2)

which we will derive shortly (at least for the portion of the manifold inside the cut locus) as a consequence of the first and second variation formulae for geodesics. (In the previous lecture, the inequality (2) was derived from a transport inequality for , but we will take a slightly different tack here.) Observe that (2) is an equality when (M,g) is a Euclidean space .

It turns out that the monotonicity of Perelman reduced volume for Ricci flows can similarly be reduced to a pointwise inequality, in which the Laplacian is replaced by a heat operator, and the radial variable r is replaced by the Perelman reduced length. More precisely, given an ancient Ricci flow for , a time , and two points , recall that the reduced length is defined as

(3)

where the -length of a curve from to is defined as

, (4)

where we adopt the shorthand , and that Conjecture 1 from the previous lecture asserts that the Perelman reduced volume

(5)

is non-increasing in for Ricci flows. If we differentiate (5) in , using the variation formula , we easily verify that the monotonicity of (5) will follow (assuming is sufficiently smooth, and that either M is compact, or grows sufficiently quickly at infinity) from the pointwise inequality

(6)

which should be viewed as a parabolic analogue to (2).

Exercise 1. Verify that (6) is an equality in the case of the (trivial) Ricci flow on Euclidean space, using Example 1 from the previous lecture. (This is of course consistent with Example 2 from that lecture.)

Exercise 2. Show that (6) is equivalent to the assertion that the function is a subsolution of the adjoint heat equation, or more precisely that . Note that this fact implies the monotonicity of Perelman reduced volume (cf. Exercise 2 from Lecture 8). [It seems that the elliptic analogue of this fact is the assertion that the Newton-type potential is subharmonic away from the origin for Ricci flat manifolds of dimension three or larger , which is a claim which is easily seen to be equivalent to (2) thanks to the Gauss lemma.]

So to prove monotonicity of the Perelman reduced volume, the main task will be to establish the pointwise inequality (6). (There are some additional technical issues, mainly concerning the parabolic counterpart of the cut locus, which we will also have to address, but we will work formally for now, and deal with these analytical matters later.)

We will perform a minor simplification: by using the rescaling symmetry (and noting the unsurprising fact that (6) is dimensionally consistent) we can normalise .

– First and second variation formulae for -geodesics –

To establish (6), we of course need some variation formulae that compute the first and second derivatives of the reduced length function . To motivate these formulae, let us first recall the more classical variation formulae that give the first and second derivatives of the metric function on a Riemannian manifold (M,g), which in particular can be used to derive (2) when the Ricci curvature is non-negative.

We recall that the distance can be defined by the energy-minimisation formula

(7)

where ranges over all curves from to x, where the Dirichlet energy of the curve is given by the formula

(8)

where we write . It is known that this infimum is always attained by some geodesic ; we shall assume this implicitly in the computations which follow.

Now suppose that we deform such a curve with respect to a real parameter , thus is now a function on the two-dimensional parameter space . The first variation here can be computed as

(9)

where is the pullback of the Levi-Civita connection on M with respect to applied in the direction ; here we of course use that g is parallel with respect to this connection. The torsion-free nature of this connection gives us the identity

(10)

where is the infinitesimal variation, and is the pullback of the Levi-Civita connection applied in the direction (cf. Exercise 5 from Lecture 6). An integration by parts (again using the parallel nature of g) then gives the first variation formula

. (11)

If we fix the endpoints of to be and , then the first term on the right-hand side of (11) vanishes. If we consider arbitrary infinitesimal variations of with fixed endpoints, we thus conclude that in order to be a minimiser for (7), that must obey the geodesic flow equation

. (12)

One consequence of this is that the speed of such a minimiser must be constant, and from (7) we then conclude

. (13)

If we then vary a geodesic with the initial endpoint fixed at and the final endpoint variable, the variation formula (11) gives

(14)

which, if we insert this back into (7) and use (13), gives

(13)

which is a (one-sided) version of the Gauss lemma. If one is inside the cut locus, then the metric function is smooth, and one can then replace the inequality with an equality by considering variations both forwards and backwards in the s variable, recovering the full Gauss lemma. In particular, we conclude in this case that is a unit vector.

Now we consider the second variation of the energy, when is already a geodesic. For simplicity we assume that evolves geodesically in the s direction, thus

. (14)

[Actually, since is already a geodesic and thus is stationary with respect to perturbations that respect the endpoints, the values of away from endpoints - which represents a second-order perturbation respecting the endpoints - will have no ultimate effect on the second variation of . Nevertheless it is convenient to assume (14) to avoid a few routine additional calculations.]

Differentiating (9) once more we obtain

. (15)

Using (10), (14), and the definition of curvature, we have

(16)

and thus (by one further application of (10))

. (17)

Now let us fix the initial endpoint and let the other endpoint vary, thus equals 0 at time t=0 and equals at time t=1. From Cauchy-Schwarz we conclude

. (18)

Actually, we can attain equality here by choosing the vector field appropriately:

Exercise 3. If we set , where v is the parallel transport of x’(s) along X, or more precisely the vector field that solves the ODE

(19)

show that all the inequalities in (18) are obeyed with equality.

For such a vector field, we conclude that

. (20)

From this formula (and the first variation formula) we conclude that

. (21)

Now let vary over an orthonormal basis of the tangent space of x(0); by (19) we see that v determines an orthonormal frame for s=0 and . Summing (21) over this basis (and using the formula for the Laplacian in normal coordinates) we conclude that

. (22)

In particular, for manifolds of non-negative Ricci curvature we have

(23)

from which (2) easily follows from the Gauss lemma. (Observe that (23) is obeyed with equality in the Euclidean case.)

Now we develop analogous variational formulae for -length (and reduced length) on a Ricci flow. We shall work formally for now, assuming that all infima are actually attained and that all quantities are as smooth as necessary for the analysis that follows to work; we then discuss later how to justify all of these assumptions. As mentioned earlier, we normalise .

Let us take a path and vary it with respect to some additional parameter s as before. Differentiating (4), we obtain

(24)

where and . On the other hand, if we have a Ricci flow , we see that

; (25)

placing this into (24) and using the fundamental theorem of calculus, we can express the right-hand side of (24) as

(26)

where G(X) is the vector field

. (27)

Here is the vector field , or equivalently it is the vector field Z such that for all vector fields W.

Note that G does not depend on Y. From this we see that in order for to be a minimiser of with the endpoints fixed, we must have G(X)=0, which is the parabolic analogue of the geodesic flow equation (14).

Example 1. In the case of the trivial Euclidean flow, the minimal -path from to takes the form where , in which case . It is not hard to verify that G=0 in this case. .

Arguing as in the elliptic case, we conclude (assuming the existence of a unique minimiser, and the local smoothness of reduced length) the first variation formula

(28)

or equivalently

. (29)

Example 2. Continuing Example 1, note that and , which is of course consistent with (28).

Having computed the spatial derivative of the reduced length, we turn to the time derivative. The simplest way to compute this is to observe that any partial segment of an -minimising path must again be a -minimising path. From (4) and the fundamental theorem of calculus we have

(30)

where we vary in by truncation; by (3) and the above discussion we conclude

(31)

where varies along (in particular, ). Applying the product and chain rules, we can expand the left-hand side of (31) as

; (32)

using (29), we conclude that

. (33)

Now we turn to the second spatial variation of the reduced length. Let be a -minimiser, so that G=0. Differentiating (24) again, we obtain

. (34)

As in the elliptic case, it is convenient to assume that we have a geodesic variation (14). In that case, we again have (16), and we also have . Using (10), we thus express (34) as

. (35)

As before, we optimise this in Y. Because the metric g now changes in time by Ricci flow, one has to modify the prescription in Exercise 3 slightly. More precisely, we now set , where v solves the following variant of (19),

. (36)

The point of doing this is that the ODE is orthogonal; the length of v is preserved along X, as is the inner product between any two such v’s (cf. equation (15) from Lecture 3). A brief computation then shows that

(37)

and hence

. (38)

Putting all of this into (35), we now see that the second variation (34) is equal to

. (39)

We now let range over an orthonormal basis of , which leads to v being an orthonormal frame at every point (0,t). Summing over (39) and also using (3), we conclude that

. (40)

Now we simplify the right-hand side of (40). The second term is of course elementary:

(41)

and this is consistent with the Euclidean case (in which is exactly when , and all curvature terms vanish). To simplify the remaining terms, we recall the variation formula

(42)

for the scalar curvature (equation (31) of Lecture 1); by the chain rule, we thu have the total derivative formula

. (43)

Inserting (41), (43) into (40) and integrating by parts, we express the right-hand side of (40) as

. (44)

To simplify this further, recall that the quantity G defined in (27) vanishes. This (and the fact that g evolves by Ricci flow ) allows one to compute the variation of :

. (45)

Inserting this into (44) and integrating by parts, one can rewrite (44) as

(46)

and so by (3) we obtain the inequality

. (47)

Combining (29), (33), and (47) we obtain (6) as desired.

– Analytical issues –

We now discuss in broad terms the analytical issues that one must address in order to make the above arguments rigorous. We first review the classical elliptic theory (i.e. the theory of geodesics in a Riemannian manifold) before turning to Perelman’s parabolic theory of -geodesics in a flow of Riemannian metrics.

In a complete Riemannian manifold, a geodesic from a fixed point to some other point has a well-defined initial velocity vector , and conversely each initial velocity vector determines a unique geodesic with an endpoint , thus defining the exponential map based at . One can show (from standard ODE theory) that this exponential map is smooth (with the derivative of this map controlled by Jacobi fields). Also, if M is connected, then any two points can be joined by a geodesic, and the exponential map is onto. However, there can be vectors v for which this map degenerates (i.e. its derivative ceases to be invertible) – these correspond to the conjugate points of in M.

Define the injectivity region of to be the set of all for which there is a unique minimising geodesic from to , and that the exponential map is not degenerate along this geodesic (in particular, and are not conjugate points). An analysis of Jacobi fields reveals that the injectivity region is open, that the distance function is smooth in this region (except at the origin), and that all the computations given above for the distance function can be justified. So it remains to understand what happens on the complement of the injectivity region, known as the cut locus. Points on the cut locus are either conjugate points to , or are else places where minimising geodesics are not unique, which (by a variant of the Gauss lemma) forces the distance function to be non-differentiable at these points. The former type of points form a set of measure zero, thanks to Sard’s theorem, whereas the latter set of points also form a set of measure zero, thanks to Radamacher’s differentiation theorem and the Lipschitz nature of the distance function (i.e. the triangle inequality). Thus the injectivity region has full measure. While this does mean that pointwise inequalities such as (2) now hold almost everywhere, this is unfortunately not quite enough to ensure that (2) holds in the sense of distributions, which is what one really needs in order to fully justify results such as the Bishop-Gromov inequality. (Indeed, by considering simple examples such as the unit circle, we see that the distribution can in fact contain some negative singular measures, although one should note that this does not actually contradict (2) due to the favourable sign of these singular components.) Fortunately, one can address this technical issue by constructing barrier functions to the radius function r at every point , i.e. functions for each which upper bound r near (and match r exactly at , and which obeys the inequality (2) at up to a loss of . Such functions can be constructed at any , even those in the cut locus, by perturbing the origin by an epsilon, and then one can use these barrier functions to justify (2) in the sense of distributions. (I believe that these arguments to control the distance function outside of the injectivity region originate with a paper of Calabi.) From this one can rigorously justify the Bishop-Gromov inequality for all radii, even those exceeding the radius of injectivity.

Analogues of the above assertions hold for the monotonicity of Perelman reduced volume on flows on compact Ricci flows (and more generally for Ricci flows of complete manifolds of bounded curvature). For instance, one can show (using compactness arguments in various weighted Sobolev spaces) that, as long as the manifold M is connected, a minimiser to (3) always exists, and is attained by an -geodesic (defined as a curve for which the G quantity defined in (27) vanishes). [One can easily reduce to the connected case, since the reduced length is clearly infinite when and lie on distinct connected components.] Such geodesics turn out to have a well-defined “initial velocity” , as can be seen by working out the ODE for the quantity (it is also convenient to reparameterise in terms of the variable to remove any apparent singularity at ). This leads to an -exponential map for any fixed time , which is smooth. The derivative of this map is controlled by -Jacobi fields, which are close analogues of their elliptic counterparts, and which lead to the notion of a -conjugate point to at the fixed time . One can then define the injectivity domain and cut locus as before (again for a fixed time ), and show as before that the former region has full measure. This lets one rigorously derive (6) almost everywhere (especially after noting that any segment of a minimising -geodesic without conjugate points is again a minimising -geodesic without conjugate points, thus establishing that the injectivity region is in some sense “star-shaped”), but again one needs to justify (6) in the sense of distributions in order to derive the monotonicity of Perelman reduced volume. This can again be done by use of barrier functions, perturbing the base point both spatially and also backwards in time by an epsilon. The details of this become rather technical; see for instance the paper of Ye or the notes of Kleiner-Lott, for details.

Thus far we have only discussed how reduced length and reduced volume behave on smooth Ricci flows of compact manifolds. Of course, to fully establish the global existence of Ricci flow with surgery, one also needs to build an analogous theory for Ricci flows with surgery. Here there turns out to be significant new technical difficulties, basically because one has to restrict attention to paths which avoid all regions in which surgery is taking place. This creates some “holes” in the region of integration for the reduced volume, as in some cases the minimising path between two points in spacetime goes through a surgery region. Fortunately it turns out that (very roughly speaking) these holes only occur when the reduced length (or a somewhat technical modification thereof) is rather large, which means that the holes do not significantly impact lower bounds on this reduced volume, which is what is needed to establish -noncollapsing. If time permits, I will discuss this issue further in later lectures, once we have described surgery in more detail.

In order to control ancient -noncollapsing solutions, which are complete but not necessarily compact, one also needs to extend the above theory to complete non-compact manifolds. It turns out that this can be done as long as one has uniform bounds on curvature; a key task here is to establish that the reduced length behaves roughly like (which is basically what it is in the Euclidean case) as goes to infinity, which allows the integrand in the definition of reduced volume to have enough decay to justify all computations. The technical details here can be found in several places, including the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.

Remark 1. A theory analogous to Perelman’s theory above was worked out earlier by Li and Yau, but with the Ricci flow replaced by a static manifold with a lower bound on Ricci curvature, and with a time-dependent potential attached to the Laplacian.

In the next few lectures we shall give a rigorous proof of this monotonicity, without using the infinite-dimensional limit and instead using results related to the Li-Yau-Hamilton Harnack inequality. (There are several other approaches to understanding Perelman’s reduced volume, such as Lott’s formulation based on optimal transport, but we will restrict attention in this course to the methods that are in Perelman’s original paper.)

– The Bishop-Gromov inequality –

Let p be a point in a complete d-dimensional Riemannian manifold . As noted in Lecture 7, we can use the exponential map to pull back M and g to the tangent space , which is also equipped with the radial variable r and the radial vector field . From Exercise 7 of Lecture 7, we have the transport equation

(1)

for the volume measure , and a transport inequality

(2)

for the Laplcian which appears in (1). In particular, if we assume the lower bound

(3)

for Ricci curvature in a ball for some real number , then from the Gauss lemma (Lemma 1 of Lecture 7) we have

. (4)

Also, from an expansion around the origin (see e.g. (13) or (15) from Lecture 7) we have

(5)

for small r. In principle, (4) and (5) lead to upper bounds on , which when combined with (1) lead to upper bounds on , which in turn lead to upper bounds on . One can of course just go ahead and compute these bounds, but one computation-free way to proceed is to introduce the model geometry , defined as

1. the standard round sphere of radius (and thus constant sectional curvature K) if (Example 1 from Lecture 7);
2. the standard hyperbolic space of constant sectional curvature K if (Example 2 from Lecture 7); or
3. the standard Euclidean space if K=0.

As all of these spaces are homogeneous (in fact, they are symmetric spaces), the choice of origin p in this model geometry is irrelevant. Observe that the orthogonal group acts isometrically on each of these spaces, with the orbits being the spheres centred at p. This implies that at any point q not equal to p, is invariant under conjugation by the stabiliser of that group on q, which easily implies that it is diagonal on the tangent space to the sphere (i.e. to the orthogonal complement of ). From this we see that for this model geometry, the inequality in (2) is in fact an equality. Since the model geometry also has constant sectional curvature K (which implies equality in (3)), we thus see that one has equality in (4) for this model geometry as well. From this we can conclude:

Lemma 1. (Relative Bishop-Gromov inequality) With the assumptions as above, the volume ratio is a non-increasing function of r as .

Exercise 1. Prove Lemma 1. (Hints: One can avoid all issues with non-injectivity by working inside the cut locus of p, which determines a star-shaped region in . In the positive curvature case K > 0, the model geometry has a finite radius of injectivity, but observe that we may without loss of generality reduce to the case when is less than or equal to that radius (or one can invoke Myers’ theorem, see Exercise 2 below). To prove the monotonicity of ratios of volumes of balls, it may be convenient to first achieve the analogous claim for ratios of volumes of spheres, and then use the Gauss lemma and the fundamental theorem of calculus to pass from spheres to balls.)

Exercise 2. Prove Myers’ theorem: if a Riemannian manifold obeys (3) everywhere for some , then the diameter of the manifold is at most . (Hint: in the model geometry, the sphere of radius r collapses to a point when r approaches .)

Remark 1. This Lemma implies the volume comparison result whenever one has bounded normalised curvature, which was used in the previous lecture; indeed, thanks to the above inequality, it suffices to prove the claim for model geometries.

Setting K=0, we obtain

Corollary 1. Let be a complete d-dimensional Riemannian manifold of non-negative Ricci curvature, and let p be a point in M. Then is a non-increasing function of r.

Let us refer to the quantity as the Bishop-Gromov reduced volume at the point p and the scale r; thus we see that this quantity is dimensionless (i.e. invariant under scaling of the manifold and of r), and non-increasing in r when one has non-negative Ricci curvature (and in particular, for Ricci-flat manifolds).

Exercise 3. Use the Bishop-Gromov inequality to state and prove a rigorous version of the following informal claim: if a Riemannian manifold is non-collapsed at a point p at one scale (as defined in Lecture 7), then it is also non-collapsed at all larger scales .

– Parabolic theory as infinite-dimensional elliptic theory –

We now come to an interesting (but still mostly heuristic) correspondence principle between elliptic theory and parabolic theory, with the latter being viewed as an infinite-dimensional limit of the former, in a manner somewhat analogous to that of the central limit theorem in probability. To get some idea of what I mean by this correspondence, consider the following (extremely incomplete, non-rigorous, inaccurate, and imprecise) dictionary:

Elliptic	Parabolic
Riemannian manifold (M,g)	Riemannian flow
Complete manifold	Ancient flow of complete manifolds
Spatial origin 0	Spacetime origin
Elliptic scaling	Parabolic scaling
Laplace equation	Heat equation
Ricci flat manifold	Ricci flow
Mean value principle	Fundamental solution
Normalised measure on the sphere	Heat kernel
Maximum principle	Maximum principle
Ball of radius O(r) around spatial origin	Cylinder of radius O(r) and height extending backwards in time from spacetime origin
Radial variable r=|x|	|x| or
Bishop-Gromov reduced volume	Perelman reduced volume

Remark 2. Of course, we have not defined Perelman reduced volume yet, but the point is that the monotonicity of Perelman reduced volume for Ricci flow is supposed to be the parabolic analogue of the monotonicity of Bishop-Gromov reduced volume for Ricci-flat manifolds. Note that one has two competing notions of the parabolic radial variable, |x| and , where is the backwards time variable; the ratio between these two competitors is essentially the Perelman reduced length, which does not really have a good analogue in the elliptic theory (except perhaps in the “latitude” variable one gets when decomposing a sphere into cylindrical coordinates).

It is well known that elliptic theory can be viewed as the static (i.e. steady state) special case of parabolic theory, but here we want to discuss a rather different connection between the two theories that goes in the opposite direction, in which we view parabolic theory as a limiting case of elliptic theory as the dimension d goes to infinity.

To motivate how this works, let us begin with a smooth ancient solution to the Euclidean heat equation

(6)

and ask how to convert it to a high-dimensional solution to the Laplace equation. At first glance this looks unreasonable: the Laplacian only contains second order derivative terms, but we have to somehow generate the first-order derivative out of this. The trick is to use polar coordinates. Recall that if we parameterise a Euclidean variable away from the origin as for and , then the Laplacian of a function can be expressed by the classical formula

(7)

where is the Laplace-Beltrami operator on the sphere. In particular, if f is a radial or spherically symmetric function (so by abuse of notation we write ), we have

. (8)

Now if we look at the high-dimensional limit (noting that f, being radial, is well defined in every dimension), we see that the first order term dominates, despite the fact that is a second order operator. To clarify this domination (and to bring into view the operator appearing in (6)), let us make the change of variables

(9)

(thus is the average of the squared coordinates ). A quick application of the chain rule then yields

(10)

(one can also see this by writing and applying the Laplacian operator directly). If we restrict attention to the region of where all the coordinates are O(1), so and , and fix while letting N go off to infinity, we thus see that converges to (with errors that are ).

Returning back to our ancient solution to the heat equation (6), it is now clear how to express this solution as a high-dimensional nearly harmonic function: if we define the high-dimensional lift of u to the N+d-dimensional Euclidean space for some large N by using the change of variables (9), i.e.

(11)

then we see from (10) and (6) that is nearly harmonic as claimed; indeed we have

(12)

in the region , which implies as before that

. (13)

Remark 3. Writing y in polar coordinates as , the metric on can be expressed as

(14)

where is the metric on the sphere of constant curvature . This expression is essentially the first equation in Section 6 of Perelman’s paper in the Euclidean case. Perelman works exclusively in polar coordinates, but I have found that the Cartesian coordinate approach can be more illuminating at times.

Remark 4. The formula (9) seems closely related to Itō’s formula from stochastic calculus, combined perhaps with the central limit theorem, though I was not able to make this connection absolutely precise. Note that for reasons of duality, stochastic calculus tends to involve the backwards heat equation rather than the forwards heat equation (see e.g. the Black-Scholes formula), which seems to explain why the minus sign in (9) is not present in Itō’s formula.

To illustrate how this correspondence could be used, let us heuristically derive the classical formula

(15)

for solutions to the heat equation (6) from the classical mean value principle

(16)

for harmonic functions . Actually, it will be slightly simpler to use the mean value principle for balls rather than spheres,

, (17)

though in high dimensions there is actually very little difference between balls and spheres (the bulk of the volume of a high-dimensional ball is concentrated near its boundary, which is a sphere).

Let and be as in (6) and (11). From (12) we see that is almost harmonic; let us be non-rigorous and pretend that is close enough to harmonic that the formula (17) remains accurate for this function. We write the volume of the ball as for some constant . As for the integrand in (17), we use polar coordinates , and rewrite (17) as

(18)

for some other constant . In view of (9), it is natural to write for some , and in view of (13) it is natural to work in the regime in which , and . Because is so rapidly increasing when N is large, the bulk of the inner integral is concentrated at its endpoint (cf. our previous remark about high-dimensional balls concentrating near their boundary), and so we expect

. (19)

Since is so much larger than |x| in our regime of interest, we can heuristically approximate by . Also, by Taylor approximation we have

(20)

Putting all this together, and substituting , we heuristically conclude

(21)

for some other constant . Taking limits as we heuristically obtain (15) up to a constant.

Exercise 4. Work through the calculations more carefully (but still heuristically), using Stirling’s approximation to the Gamma function, together with the classical formulae for the volume of balls and spheres, to verify that one does indeed get the right constant of in (15) at the end of the day (as one must).

Now let us perform a variant of the above computations which is more closely related to the monotonicity of Perelman’s reduced volume. The Euclidean space is of course Ricci-flat, and so from Corollary 1 we know that the Bishop-Gromov reduced volume

(22)

is non-decreasing in (and thus non-decreasing in ). (Of course, being Euclidean, (22) is equal to a constant ; but let us ignore this fact (which we have already used in our heuristic derivation of (15)) for now.) Repeating all the above computations (but with u and replaced by 1) we thus heuristically conclude that the quantity

(23)

is also non-decreasing in . (Indeed, this quantity is equal to for all .) The quantity (23) is precisely the Perelman reduced volume of Euclidean space (which we view as a trivial example of an ancient Ricci flow) at the spacetime origin (0,0) and backwards time parameter .

– From Ricci flow to Ricci flat manifolds –

We have seen how ancient solutions to the heat equation on a Euclidean spacetime can be viewed as (approximately) harmonic functions on an “infinitely high dimensional” Euclidean space. Now we would like to analogously view ancient solutions to a heat equation on a flow of Riemannian manifolds as harmonic functions on an “infinitely high dimensional” Riemannian manifold, and similarly to view ancient Ricci flows as infinite dimensional infinitely high dimensional Ricci-flat manifolds.

Let’s begin with the former task. Starting with an ancient flow of d-dimensional Riemannian metrics for (which we will not assume to be a Ricci flow just yet) and a large integer N, we can consider the N+d-dimesional manifold . As a first attempt to mimic the situation in the Euclidean case, it is natural to endow with the Riemannian metric given by the formula

(24)

where t is given by the formula (9). In terms of local coordinates, if we use the indices a,b,c to denote the d indices for the x variable and i,j,k to denote the N indices for the y variable, we have

(25)

where is the Kronecker delta. From this we see that the volume measure on is given by

(26)

and the Dirichlet form

(27)

for this Riemannian manifold is given by

, (28)

where is the gradient of u in the x variable using the metric g(t). We can then integrate by parts to compute the Laplacian . Recalling that varies in t by the formula

(29)

(equation (19) from Lecture 1) and using (9) and the chain rule, we see that

(30)

where is the Laplace-Beltrami operator in the x variable using the metric g(t). If we specialise to radial functions

(31)

and use (10) and the chain rule, we can rewrite (29) as

(32)

Thus we see that if u solves the heat equation , then its lift is approximately harmonic in the sense that in the region where and x is confined to a compact region of space.

Remark 5. The term in (32) is somewhat annoying; we will later tweak the metric (24) in order to remove it (at the cost of other, more acceptable, terms).

Now let us see whether Ricci flows lift to approximately Ricci-flat manifolds . We begin by computing the Christoffel symbols in local coordinates, where refer to the N+d combined indices coming from the indices a on M and the indices i on . We recall the standard formula

(33)

for the Christoffel symbols of a general Riemannian manifold in local coordinates. Specialising to the metric (25), some computation reveals that

(34)

.

Now the Ricci curvature can be computed from the Christoffel symbols by the standard formula

. (35)

If we apply this formula we obtain (after some computation)

(36)

.

We thus see that if the original flow obeys the Ricci flow equation , then the lifted manifold is nearly Ricci flat in the sense that all components of the Ricci curvature tensor are O(1/N) (in the region ). In fact the above estimates show that the Ricci curvature tensor is also O(1/N) in the operator norm sense and in the Hilbert-Schmidt (or Frobenius) sense.

It turns out that this approximation is not quite good enough for applications to Ricci flow, mainly because the term in (36) gives a significant contribution to the trace of the Ricci tensor (i.e. the scalar curvature ), even in the limit . It turns out however that one can eliminate this problem by adding a correction term to the metric (24) involving the scalar curvature. More precisely, given an ancient Ricci flow , define the modified metric by the formula

(37)

where of course is the derivative of the radial variable r, and R(t,x) is the scalar curvature of g(t) at x. In coordinates, we have

. (38)

Exercise 5. Let be a smooth ancient Ricci flow on , and let be defined by (37). Show that in the region where (so ) and x ranges in a compact set, the Christoffel symbols take the form

(39)

.

and the Ricci curvature takes the form

(40)

.

In particular, has norm in the trace (i.e. nuclear) norm (and hence in the Hilbert-Schmidt/Frobenius and operator norms).

Exercise 6. Let the assumptions and notation be as in Exercise 5, let be a smooth function, and let be as in (31). Show that the Laplacian associated to obeys a similar formula to (32), but with the term replaced by terms which are when t, x are bounded.

– Perelman’s reduced length and reduced volume –

In the previous discussion, we have converted a Ricci flow to a Riemannian manifold of much higher dimension which is almost Ricci flat. Let us adopt the heuristic that this latter manifold is sufficiently close to being Ricci flat that the Bishop-Gromov inequality (Corollary 1) holds (at least in the asymptotic limit ), thus the Bishop-Gromov reduced volume should heuristically be non-increasing in , where we fix a spatial origin .

In order to exploit the above heuristic, we first need to understand the distance function on . Let be a point in , and consider a length-minimising geodesic from to , where we have normalised the length of the parameter interval by the formula .

Observe that the metric (37) can be rewritten in polar coordinates (after substituting ) as

(41)

(which is essentially the first formula in Section 6 of Perelman’s paper). Note that the angular variable only influences the second term in this metric and not the other two. Because of this, one sees that the geodesic must keep constant in order to be length-minimising (i.e. for the duration of the geodesic). Turning next to the variable, we then see that for N large enough, the geodesic should increase continuously from 0 to (as the term in (41) will severely penalise any backtracking. After a reparameterisation we may in fact assume that increases at constant speed, thus we have

(42)

for some path from to . Using (41), the length of this geodesic is

(43)

which by Taylor expansion is equal to

(44)

where the -length of is defined as

(45)

Note that this quantity is independent of N. Thus, heuristically, geodesics in from to should (approximately) minimise the -length. If we define to be the infimum of over all paths from to , we thus obtain the heuristic approximation

. (46)

Exercise 7. When M is the Euclidean space (with the trivial Ricci flow, of course), show that , and the minimiser is given by .

From (46) we see that the ball in of radius centred at (where, as always, we are in the regime , so ) should heuristically take the form

. (47)

If we make the plausible assumption that varies smoothly in , then (47) is heuristically close (when N is large) to

(48)

or equivalently

. (49)

Now, the volume measure of (37) is of the form , and so the volume of (49) is approximately

. (50)

(Note there is a slight abuse of notation since depends on , but it will soon be clear that this abuse is harmless.) When N is large, the inner integral is dominated by its right endpoint as before, and so (50) is approximately

. (51)

We can Taylor expand this to be approximately

(52)

where the Perelman reduced length is defined as

(53)

Example 1. Continuing the Euclidean example of Exercise 7, we have , which is the familiar exponent in the fundamental solution (15). This is, of course, not a coincidence.

From (52) we thus heuristically conclude that the Bishop-Gromov reduced volume of at and at radius is approximately equal to a constant multiple of , where the Perelman reduced volume is defined as

. (54)

Example 2. Again continuing the Euclidean example, the reduced volume in Euclidean space (with the trivial Ricci flow) is always .

Formally applying Corollary 1, we are thus led to

Conjecture 1. (Monotonicity of Perelman reduced volume) Let be a Ricci flow on , and let . Then the quantity for is monotone non-increasing in .

Remark 6. Note here we are not taking the Ricci flow to be ancient; this would correspond to the manifold being replaced by an incomplete manifold, of radius about . However, because of the restriction , the above heuristic arguments never “encounter” the lack of completeness, and so it is reasonable to expect that the conjecture will continue to hold in the non-ancient case. This is of course an essential point for our applications, since the Ricci flows we study are not assumed to be ancient.

Remark 7. At an crude heuristic level, the Perelman reduced volume is roughly like (since, in view of Exercise 7, we expect to behave like , especially in regions of bounded normalised curvature, where we are deliberately vague about exactly what metric we using to define d). This heuristic suggests that Conjecture 1 should be able to establish the non-collapsing result we want (Theorem 2 from Lecture 7). This will be made more rigorous in subsequent lectures. For now, we observe that the Perelman reduced length and reduced volume are dimensionless (just as the Bishop-Gromov reduced volume is), which as discussed in Lecture 7 is basically a necessary condition in order for this quantity to force non-collapsing of the geometry.

As far as I am aware, there is no rigorous proof of Conjecture 1 that follows the above high-dimensional comparison geometry heuristic argument. Nevertheless, it is possible to prove Conjecture 1 by other means, and in particular by developing parabolic analogues of all the comparison geometry machinery that is used to prove the Bishop-Gromov inequality (and in particular, developing a theory of -geodesics analogous to the “elliptic” theory of geodesics on a Riemannian manifold. This will be the focus of the next few lectures.

Remark 8. It seems of interest to try to make the above arguments more rigorous, and to expand the dictionary between elliptic and parabolic equations. I do not know however of much literature in this direction, apart from Section 6 of Perelman’s original paper (see also Section 3.1 of Cao-Zhu), in which a few other parabolic notions (e.g. the backwards heat equation, or the modified Ricci flow from the previous lecture) are reinterpreted as high-dimensional elliptic notions. See however the work of Chow and Chu (see also this sequel paper), which views parabolic theory as a degenerate version of elliptic theory; Perelman’s viewpoint can be interpreted as a regularisation of Chow-Chu’s viewpoint.

(1)

on a compact Riemannian manifold (M,g) (with metric g static, i.e. independent of time), where is a scalar field, can be interpreted as the gradient flow for the Dirichlet energy functional

(2)

using the inner product associated to the volume measure . Indeed, if we evolve f in time at some arbitrary rate , a simple application of integration by parts (equation (29) from Lecture 1) gives

(3)

from which we see that (1) is indeed the gradient flow for (3) with respect to the inner product. In particular, if f solves the heat equation (1), we see that the Dirichlet energy is decreasing in time:

. (4)

Thus we see that by representing the PDE (1) as a gradient flow, we automatically gain a controlled quantity of the evolution, namely the energy functional that is generating the gradient flow. This representation also strongly suggests (though does not quite prove) that solutions of (1) should eventually converge to stationary points of the Dirichlet energy (2), which by (3) are just the harmonic functions (i.e. the functions f with ).

As one very quick application of the gradient flow interpretation, we can assert that the only periodic (or “breather”) solutions to the heat equation (1) are the harmonic functions (which, in fact, must be constant if M is compact, thanks to the maximum principle). Indeed, if a solution f was periodic, then the monotone functional E must be constant, which by (4) implies that f is harmonic as claimed.

It would therefore be desirable to represent Ricci flow as a gradient flow also, in order to gain a new controlled quantity, and also to gain some hints as to what the asymptotic behaviour of Ricci flows should be. It turns out that one cannot quite do this directly (there is an obstruction caused by gradient steady solitons, of which we shall say more later); but Perelman nevertheless observed that one can interpret Ricci flow as gradient flow if one first quotients out the diffeomorphism invariance of the flow. In fact, there are infinitely many such gradient flow interpretations available. This fact already allows one to rule out “breather” solutions to Ricci flow, and also reveals some information about how Poincaré’s inequality deforms under this flow.

The energy functionals associated to the above interpretations are subcritical (in fact, they are much like ) but they are not coercive; Poincaré’s inequality holds both in collapsed and non-collapsed geometries, and so these functionals are not excluding the former. However, Perelman discovered a perturbation of these functionals associated to a deeper inequality, the log-Sobolev inequality (first introduced by Gross in Euclidean space). This inequality is sensitive to volume collapsing at a given scale. Furthermore, by optimising over the scale parameter, the controlled quantity (now known as the Perelman entropy) becomes scale-invariant and prevents collapsing at any scale – precisely what is needed to carry out the first phase of the strategy outlined in the previous lecture to establish global existence of Ricci flow with surgery.

The material here is loosely based on Perelman’s paper, Kleiner-Lott’s notes, and Müller’s book.

– Ricci flow as gradient flow –

We would like to represent Ricci flow

(5)

as a gradient flow of some functional (with respect to some inner product, or at least with respect to some Riemannian metric on the space of all metrics g). We will assume that all quantities are smooth and that the manifold is either compact or that all expressions being integrated are rapidly decreasing at infinity (so no boundary terms etc. arise from integration by parts).

To do this, our starting point will be the first variation formula for the scalar curvature R (equation (15) from Lecture 1) for an arbitrary instantaneous deformation of the metric g:

. (6)

We can integrate in M to eliminate the latter two terms on the right-hand side (by Stokes theorem, see equation (28) from Lecture 1) to get

. (7)

This looks rather promising; it suggests that if we introduce the Einstein-Hilbert functional

(8)

then the Ricci flow (5) might be interpretable as a gradient flow for -2H.

Unfortunately, there is a problem because R is not the only time-dependent quantity in the right-hand side of (8); the volume measure also evolves in time by the formula

(9)

(see equation (19) from Lecture 1). Thus, from the product rule, the true variation of the Einstein-Hilbert functional is given by the formula

(10)

So the gradient flow of -2H (using the inner product associated to ) is not Ricci flow, but is instead a rather strange flow

(11)

where is the Einstein tensor. This flow does not have any particularly nice properties in general (it is not parabolic in three and higher dimensions, even after applying the de Turck trick from Lecture 1). On the other hand, in two dimensions the right-hand side of (10) vanishes and H(M,g) becomes invariant under deformations (we have already exploited this fact to prove the Gauss-Bonnet formula, see Proposition 1 from Lecture 4). More generally, we recover see from (10) the fact (well known in general relativity) that the (formal) stationary points of the Einstein-Hilbert functional are precisely the solutions of the vacuum Einstein equations (or equivalently, in any dimension other than 2).

We see that the variation of the measure in time is causing us some difficulty. To fix this problem, let us take the (rather non-geometric looking) step of replacing this evolving measure by some static measure which we select in advance, and consider instead the variation of the functional with respect to some arbitrary perturbation . Now that m is static, we can apply (6) to get

. (12)

Previously, we used Stokes’ theorem to eliminate the latter two terms on the right-hand side to leave us with the one term that we do want. Unfortunately, Stokes’ theorem only applies for the volume measure , not for our static measure ! In order to apply Stokes’ theorem, we must therefore convert the static measure back to volume measure. The Radon-Nikodym derivative of the two measures should be some positive function, which we shall denote by for some scalar (and time-varying) function , thus

. (13)

Inserting (13) into (12), integrating by parts using the volume measure , and then using (13) again to convert back to the static measure , we see after a little calculation that

(14)

and similarly

(15)

and so we can express the right-hand side of (12) as

. (16)

This looks rather unpleasant; we managed to eradicate the scalar curvature term that was present in the variation in (10), but at the cost of introducing four new terms involving f. But to deal with this, first observe from differentiating (13) and using (9) and the static nature of dm that we know the first variation of f:

. (17)

So the term that appears in (16) can be rewritten as . Now this term looks familiar… in fact, it essentially the variation (3) of the Dirichlet energy functional for the measure dm! This suggests that we may be able to simplify (16) if we modify our functional by adding some multiple of the Dirichlet functional .

One cannot apply (3) directly, though, because (a) g is evolving in time, rather than static, and also (b) dm is not the volume measure for g. But we have all the equations to deal with this, and one can compute the first variation of E:

Exercise 1. Show that

. (18)

(Hint: expand out and use (3) from Lecture 1.)

If we thus define the functional

(19)

we see from (16), (18) that we get a lot of cancellation, ending up with

. (20)

Thus the gradient flow of with respect to the inner product on symmetric two-forms (or more precisely, on the tangent space of such forms at g) is given by

. (21)

From (17) we see that f now evolves by a backward heat equation

. (22)

With this flow, we see that is monotone increasing, with

. (23)

The equation (21) is almost Ricci flow (5), but with one additional term associated with f. But we can observe (using equation (25) from Lecture 1) that is just the Lie derivative of g in the direction of the gradient vector field . Thus we see that (23) is a modified Ricci flow (see equation (36) from Lecture 1), which is conjugate to genuine Ricci flow by a diffeomorphism as discussed in that lecture. Thus while we have not established Ricci flow as a gradient flow directly, we have managed to find a whole family of gradient flows (parameterised by a choice of static measure dm, or equivalently by a choice of potential function f evolving by (17)) which are equivalent to Ricci flow modulo diffeomorphism. (Indeed, by placing an appropriate Riemannian structure on the moduli space of metrics modulo diffeomorphism, one can express Ricci flow modulo diffeomorphism as a true (formal) gradient flow; see Section 9 of the Kleiner-Lott notes.) As remarked in Perelman’s paper, one can view f as a kind of gauge function for the Ricci flow.

Example 1. If (M,g) is a Euclidean space with the contracted Euclidean metric for times , where and is the standard metric, with equal to the Gaussian measure (thus ), then g, f solve (21), (22). (One has to be a bit careful here because M is non-compact, of course.)

We can of course conjugate away the infinitesimal diffeomorphism given by the vector field , which converts the system (21), (22) to the system

(24)

(here we use the fact that ), which is Ricci flow coupled with a nonlinear backwards heat equation for the potential f. (Note that the equation for f is not always solvable forwards in time for any non-zero amount of time, but we can always solve it instantaneously for a fixed time, which is good enough for first variation analysis.) The non-linear backwards heat equation equation for f can be linearised by setting , in which case it becomes the adjoint heat equation

. (24′)

Exercise 2. Writing , show that (24′) is equivalent to the equation

(24”)

where is viewed as a d-form for the purposes of applying the Laplacian. Thus the adjoint heat equation can be viewed as the backwards heat equation for d-forms.

Example 2. If (M,g) is a static Euclidean space and with and the time variable t is restricted to be less than , then g, f solve (24), and is the Gaussian measure , which solves the backwards heat equation. Note that this is the conjugated version of Example 1. Again, one needs to take care because M is non-compact.

By performing this conjugation, the measure m is no longer static, and we reflect this by changing the notation a little to

. (25)

The relationship between and the flow (24) is analogous to that between and (21), (22). For instance, we have the following analogue of (23):

Exercise 3. If g, f solve (24), show that

. (26)

Thus is monotone non-decreasing in time. We would like to use this to develop a controlled quantity for Ricci flow, but we need to eliminate f. This can be accomplished by taking an infimum, defining

. (27)

The normalisation (which makes dm a probability measure) is needed to ensure a meaningful infimum; note that this normalisation is preserved by the flow (24) since dm is only moved around by diffeomorphisms. This quantity has an interpretation as the best constant in a Poincaré inequality:

Exercise 4. Show that is the least number for which one has the inequality

(28)

for all in the Sobolev space . (Hint: reduce to the case when u is positive and smooth and then make the substitution .) Conclude in particular that is finite, that it is the least eigenvalue of the self-adjoint modified Laplacian , and lies between and the average scalar curvature .

A variational argument (using the standard fact that embeds compactly into ) shows that equality in (28) is attained by some strictly positive with norm , and so the infimum in (27) is also attained for some f. Applying the flow (24) instantaneously at a given time, we conclude (formally, at least) that we have the monotonicity formula

(29)

for any solution to Ricci flow (5), where f is the extremiser for (27) (note that this extremiser f need not evolve via (25)). (One can in fact make this formula rigorous whenever the Ricci flow is smooth and M is compact, but we will not detail this here.)

This monotonicity is similar to the monotonicity of . For instance, the functional has a dimension of -2 in the sense of the previous lecture, same as . As further evidence of similarity, we have:

Exercise 5. Show that , and use this to conclude an analogue of Proposition 1 from Lecture 3 for . In particular conclude that Ricci flow must develop a finite time singularity if is positive.

Exercise 6. If is a Ricci flow which is a steady breather in the sense that it is periodic modulo isometries (thus is isometries to for some ), show that at time zero we have

(30)

for some . Conclude that , thus simply evolves by diffeomorphism by the gradient field f. (For this you may need to use the uniqueness of the initial value problem for Ricci flow.) In other words, all steady breathers are gradient steady solitons.

Remark 1. One can apply a similar argument to deal with compact expanding breathers (in which is isometric to a larger dilate of for some by normalising by a power of the volume as in Exercise 1 of Lecture 7, concluding that such breathers are necessarily gradient expanding solitons with

(31)

at time zero for some potential f and some ; see Perelman’s paper (or Section 7 of Kleiner-Lott) for details. With a little more work (using the maximum principle) one can in fact show that f is constant, and so the only compact expanding breathers are Einstein manifolds. (Actually, this result can also be established using Exercise 1 from Lecture 7 directly, as follows from the work of Hamilton.) This normalisation of is also closely related to the Yamabe invariant of M; see this paper of Kotschick for further discussion.

Example 3. Any Ricci-flat manifold (i.e. ) is of course a gradient steady soliton with . A more non-trivial example is given by Hamilton’s cigar soliton (also known as Witten’s black hole), which is the two-dimensional manifold with the conformal metric and gradient function ; we leave the verification of the gradient shrinking property (30) as an exercise.

Remark 2. If Ricci flow was a gradient flow for a functional which was geometric (or more precisely, invariant under diffeomorphism), then this flow could not deform a metric by any non-trivial diffeomorphism (since this is a stationary direction for this functional, rather than a steepest descent). Thus the existence of non-trivial gradient steady solitons, such as the cigar soliton, explains why Ricci flow cannot be directly expressed as a gradient flow without introducing a non-geometric object such as the reference measure dm or the potential function f. (See also Proposition 1.7 of Müller’s book for a different way of seeing that Ricci flow is not a pure gradient flow.)

Exercise 7. If (M,g) is a gradient steady soliton with potential f, show that , , and . (Hint: to prove the second identity, differentiate (30) and use the second Bianchi identity (Exercise 7 from Lecture 0).) Use the maximum principle to then conclude that the only compact gradient steady solitons are the Ricci-flat manifolds.

– Nash entropy –

Let us return to our analysis of the functional , in which was fixed and g evolved by the modified Ricci flow (21) (which forced f to evolve by the backwards heat equation (22)). We then obtained the monotonicity formula (23). We shall normalise dm to be a probability measure.

We can squeeze a little bit more out of this formula – in particular, making it scale invariant – by introducing the Nash entropy

(31)

which is the relative entropy of with respect to the background measure dm. (Some further relations and analogies between the functionals described here and notions of entropy from statistical mechanics are discussed in Perelman’s paper.) From (22) and one integration by parts (using (13), of course) we know how this entropy changes with time:

. (32)

To exploit this identity, let us first consider the case of gradient shrinking solitons:

Exercise 8. Suppose that a Riemannian manifold (M,g)=(M,g(0)) verifies an equation of the form

(33)

for some function f and some . Show that this equation is preserved for times if g evolves by Ricci flow, if evolves by (i.e. ), and , and that for all . Such solutions are known as gradient shrinking solitons; they combine Ricci flow with the diffeomorphism and scaling flows from Lecture 1. Note that any positively curved Einstein manifold, such as the sphere, will be a gradient shrinking soliton (with f=0). Example 1 also shows that Euclidean space can also be viewed as a gradient shrinking soliton.

If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the gradient shrinking solitons. In analogy with (23), we would therefore like the variation of this monotone quantity with respect to Ricci flow to look something like

(34)

where is some quantity decreasing at the constant rate

. (35)

But the scaling is wrong; time has dimension 2 with respect to the Ricci flow scaling, and so the dimension of a variation of a scale-invariant quantity should be -2, while the expression (34) has dimension -4. (Note that f should be dimensionless (up to logarithms), has the same dimension of time, i.e. 2, and is of course dimensionless.) So actually we should be looking at

. (36)

To find a functional whose derivative is (36), we expand the integrand as

. (37)

Using (32) and the normalisation , we can thus express (36) as

. (38)

Using (32) and (35), we can express this as a total derivative:

. (39)

Thus the quantity in parentheses is monotone increasing in time under Ricci flow (and with f, evolving by (22), (35)).

In analogy with Example 1, we rewrite the potential function f as

(40)

then obeys the slight variant of (22)

(41)

and is related to the fixed measure m by the formula

(42)

and the equality between (36) and (39) becomes

(43)

where

. (44)

The -d term here is harmless (since m is fixed), and is in place to normalise this expression to vanish in the Euclidean case (Example 1, where now ).

As before, it is convenient to conjugate away the diffeomorphism by to recover a pure Ricci flow. Define the Perelman entropy of a manifold , a scalar function , and a positive real , by

. (45)

Note that this quantity has dimension 0 (if f is viewed as dimensionless, and given the dimension 2).

Exercise 9. Suppose that g evolves by Ricci flow (5), f evolves by the nonlinear backward heat equation

, (46)

and evolves by (35). Show that

. (47)

If we write , show that (46) is also equivalent to the adjoint heat equation

. (48)

We have thus obtained a scale-invariant monotonicity formula, albeit one which depends on two additional time-varying parameters, f and . To eliminate them, the obvious thing to do is to just take the infimum over all f and ; but we need to be sure that the infimum exists at all. This will be studied next.

– Connection to the log-Sobolev inequality –

We have just established the monotonicity formula (47) whenever g evolves by Ricci flow (5) and f, evolve by (46), (35). Let us now temporarily specialise to the case when is a static Euclidean space (which of course obeys Ricci flow), and (which of course obeys (35)), and now restrict to negative times . Now all curvatures vanish, thus for instance by (48) we see that obeys the free backwards heat equation . We will normalise to be a probability measure, thus .

Example 4. The key example to keep in mind here is , in which case u becomes the backwards heat kernel .

We can now re-express the functional (45) in terms of u as

. (49)

One easily verifies by direct calculation that this expression vanishes in the model case of Example 4. For more general u, we know that this quantity is monotone increasing in time, and so

. (50)

Now suppose u is some non-negative test function at time zero with total mass 1, then from the fundamental solution for the backwards heat equation we have

(51)

where is the renormalised solution

. (52)

Observe that converges pointwise to as for fixed x. Thus in some renormalised sense this general solution is converging to the model solution in Example 3 in the limit .

We can rewrite the functional (49) after some calculation as

. (53)

One can check that is converging pointwise to . A careful application of dominated convergence then shows that in the limit , (53) converges to the value attained in Example 3, i.e. zero. By the monotonicity formula, we have thus demonstrated that

(54)

for all times . Writing and rearranging (49), we conclude the log-Sobolev inequality

(55)

valid whenever and .

Exercise 10. By letting be standard Gaussian measure and writing , deduce the original log-Sobolev inequality

(56)

of Gross, valid whenever . [One key feature of this inequality, as compared to more traditional Sobolev inequalities, is that it is almost completely independent of the dimension d.]

Remark 3. We have seen how knowledge of the heat kernel can lead to log-Sobolev inequalities, by evolving by the (backwards) heat flow (this is an example of the semigroup method for proving inequalities). This connection can in fact be reversed, using log-Sobolev inequalities to deduce information about heat kernels. Heat kernels can in turn be used to deduce ordinary Sobolev estimates, which then imply log-Sobolev estimates by convexity inequalities such as Hölder’s inequality, thus showing that all these phenomena are morally equivalent. There is a vast literature on these subjects (and other related topics, such as hypercontractivity); so much so that there are not only multiple surveys on the subject, but even a survey of all the surveys.

We now return to the case of general Ricci flows (not just the Euclidean one).

Exercise 11. Let (M,g) be a compact Riemannian manifold, and let . Using the Euclidean log-Sobolev inequality (48), show that we have a lower bound of the form for all functions f with . Show in fact that can be chosen to depend only on , the dimension, an upper bound for the magnitude of the RIemann curvature, and a lower bound for the injectivity radius. Using a rescaling and compactness argument, show also that we can take as . (Details can be found in Section 3.1 of Perelman’s paper.)

We can now define the quantity to be the infimum of for all functions f with ; thus is non-decreasing if we evolve by (35). Thus we have obtained a one-parameter family of dimensionless monotone quantities (recalling that has dimension 2 with respect to scaling).

Remark 4. One can interpret as a nonlinear analogue of the eigenvalue . Indeed, just as is the least number for which one can solve the linear eigenfunction equation

(57)

subject to the constraint , is the least number for which one can solve the nonlinear eigenfunction equation

(58)

subject to the constraints and . In particular we expect to behave roughly like in the limit .

Exercise 12. Show that the only shrinking breathers (those in which is isometric to a contraction of for some ) are the gradient shrinking solitons.

– Non-collapsing –

We now relate log-Sobolev inequalities (i.e. lower bounds on ) to non-collapsing. We first note that by substituting into (45) as in the Euclidean case, that we have the log-Sobolev inequality

(59)

whenever is non-negative with .

To use this, suppose we have a ball which has bounded normalised curvature, so in particular on this ball.
On the other hand, if is supported on B with mass 1, then from Jensen’s inequality we have

(54)

and we thus conclude from (59) that

. (60)

If we let , where is a bump function that equals 1on [-1/2,1/2] and is supported on [-1,1] (thus on the ball , and is the normalisation constant needed to ensure that has mass one, then on this ball, and so we conclude

. (61)

At this point we need to invoke the relative Bishop-Gromov inequality from comparison geometry, which among other things ensures that under the assumption of bounded normalised curvature. Indeed, from equations (15) and (17) from the previous lecture we see that inside the ball of radius , from which the claim easily follows within the radius of injectivity. (To generalise the inequality beyond this region, one simply works on the region inside the cut locus, which is star-shaped around the origin in .)

Using this inequality, we thus conclude that

. (62)

Thus a lower bound on enforces non-collapsing of volume at scale .

Exercise 13. Use (62), Exercise 11 and the monotonicity properties of to establish -noncollapsing of Ricci flows (Theorem 2 from the previous lecture).

Remark 5. This argument in fact establishes a stronger form of non-collapsing, in which in order to get non-collapsing at time and scale , one only needs bounded normalised curvature at time (instead of on the time interval ). It also works in arbitrary dimension. The second proof of non-collapsing that we will give, based on the Perelman reduced volume instead of Perelman entropy, needs the spacetime bounded normalised curvature assumption but also works in arbitrary dimension.

Remark 6. The parameter in the above result, which measures the quality of the non-collapsing, will deteriorate with time T. This is because the decay of from (35) entails that in order to get non-collapsing of the manifold at time and scale , one needs some non-collapsing at time zero and scale . Of course, since the manifold is initially compact, one always has some non-collapsing at each scale, but the quantitative constants associated to this non-collapsing will deteriorate as the scale increases, which will happen when T increases. Fortunately (and especially in view of our finite time extinction results) we only need to analyse Ricci flow on compact (though potentially rather large) time intervals .

Remark 7. It was recently shown by Zhang that the monotonicity properties of the quantities also hold for Ricci flows with surgery. This should enable one to completely replace all applications of Perelman reduced volume in the existing proof of the Poincaré conjecture in the literature by Perelman (as well as in the expositions of Kleiner-Lott, Cao-Zhu, and Morgan-Tian) by Perelman entropy, which may lead to a shorter proof overall (although one still needs the Perelman reduced length for another purpose, namely to control the geometry of ancient non-collapsed Ricci flows). However, we shall mostly follow the original arguments of Perelman in this course.

Remark 8. The above entropy functionals are also useful for studying the forward or backward heat equation on a static Riemannian manifold (basically, one keeps the heat-type equations for u or f but now replace Ricci flow by the trivial flow ). However, some sign assumptions on curvature are now needed to recover the same type of monotonicity results. See this paper of Ni for details.

[Update, April 25: some corrections.]

As mentioned in Lecture 1, local existence of the Ricci flow is a fairly standard application of nonlinear parabolic theory, once one uses de Turck’s trick to transform Ricci flow into an explicitly parabolic equation. The trouble is, of course, that Ricci flow can and does develop singularities (indeed, we have just spent several lectures showing that singularities must inevitably develop when certain topological hypotheses (e.g. simple connectedness) or geometric hypotheses (e.g. positive scalar curvature) occur). In principle, one can use surgery to remove the most singular parts of the manifold at every singularity time and then restart the Ricci flow, but in order to do this one needs some rather precise control on the geometry and topology of these singular regions. (In particular, there are some hypothetical bad singularity scenarios which cannot be easily removed by surgery, due to topological obstructions; a major difficulty in the Perelman program is to show that such scenarios in fact cannot occur in a Ricci flow.)

In order to analyse these singularities, Hamilton and then Perelman employed the standard nonlinear PDE technique of “blowing up” the singularity using the scaling symmetry, and then exploiting as much “compactness” as is available in order to extract an “asymptotic profile” of that singularity from a sequence of such blowups, which had better properties than the original Ricci flow. [The PDE notion of a blowing up a solution around a singularity, by the way, is vaguely analogous to the algebraic geometry notion of blowing up a variety around a singularity, though the two notions are certainly not identical.] A sufficiently good classification of all the possible asymptotic profiles will, in principle, lead to enough structural properties on general singularities to Ricci flow that one can see how to perform surgery in a manner which controls both the geometry and the topology.

However, in order to carry out this program it is necessary to obtain geometric control on the Ricci flow which does not deteriorate when one blows up the solution; in the jargon of nonlinear PDE, we need to obtain bounds on some quantity which is both coercive (it bounds the geometry) and either critical (it is essentially invariant under rescaling) or subcritical (it becomes more powerful when one blows up the solution) with respect to the scaling symmetry. The discovery of controlled quantities for Ricci flow which were simultaneously coercive and critical was Perelman’s first major breakthrough in the subject (previously known controlled quantities were either supercritical or only partially coercive); it made it possible, at least in principle, to analyse general singularities of Ricci flow and thus to begin the surgery program discussed above. (In contrast, the main reason why questions such as Navier-Stokes global regularity are so difficult is that no controlled quantity which is both coercive and critical or subcritical is known.) The mere existence of such a quantity does not by any means establish global existence of Ricci flow with surgery immediately, but it does give one a non-trivial starting point from which one can hope to make progress.

To be a more precise, recall from Lecture 1 that the Ricci flow equation , in any spatial dimension d, has two basic symmetries (besides the geometric symmetry of diffeomorphism invariance); it has the obvious time-translation symmetry (keeping the manifold M fixed), but it also has the scaling symmetry

(1)

for any (again keeping M fixed as a topological manifold). When applied with , this scaling shrinks all lengths on the manifold M by a factor (recall that the length of a tangent vector v is given by the square root of ), and also speeds up the flow of time by a factor ; conversely, when applied with , the scaling expands all lengths by a factor , and slows down the flow of time by .

Suppose now that one has a Ricci flow which becomes singular at some time T > 0. To analyse the behaviour of the flow as one approaches the singular time T, one picks a sequence of times approaching T from below, a sequence of marked points on the manifold, and a sequence of length scales which go to zero as . One then considers the blown up Ricci flows , where is equal to M as a topological manifold (with as a marked point or “origin” O), and is the flow of metrics given by the formula

. (2)

Thus the flow represents a renormalised flow in which the time has been redesignated as the temporal origin 0, the point has been redesignated as the spatial origin O, and the length scale has been redesignated as the unit length scale (and the time scale
has been redesignated as the unit time scale). Thus the behaviour of the rescaled flow at unit scales of space and time around the spacetime origin (thus and ) correspond to the behaviour of the original flow at spatial scale and time scale around the spacetime point , thus and .

Because the original Ricci flow existed on the time interval , the rescaled Ricci flow will exist on the time interval . In particular, in the limit (leaving aside for the moment the question of what “limit” means precisely here), these Ricci flows become increasingly ancient, in that they will have existed on the entire past time interval in the limit.

The strategy is now to show that these renormalised Ricci flows (with the marked origin O) exhibit enough “compactness” that there exists a subsequence of such flows which converge to some asymptotic limiting profile in some sense. (We will define the precise notion of convergence of such flows later, but pointed Gromov-Hausdorff convergence is a good first approximation of the convergence concept to keep in mind for now.) If the notion of convergence is strong enough, then we will be able to conclude that this limiting profile of Ricci flows is also a Ricci flow. (Actually, due to the parabolic smoothing effects of Ricci flow, we will be able to automatically upgrade weak notions of convergence to strong ones, and so this step is in fact rather easy.) This limiting Ricci flow has better properties than the renormalised flows; for instance, while the renormalised flows are almost ancient, the limiting flow actually is an ancient solution. Also, while the Hamilton-Ivey pinching phenomenon from Lecture 3 suggests that the renormalised flows have mostly non-negative curvature, the limiting flow will have everywhere non-negative curvature (provided that the points and scales are chosen properly; we will return to this “point-picking” issue later in this course).

If one was able to classify all possible asymptotic profiles to Ricci flow, this would yield quite a bit of information on singularities to such flows, by the standard and general nonlinear PDE method of compactness and contradiction. This method, roughly speaking, runs as follows. Suppose we want to claim that whenever one is sufficiently close to a singularity, some scale-invariant property P eventually occurs. (In our specific application, P is roughly speaking going to assert that the geometry and topology of high-curvature regions can be classified as belonging to one of a short list of possible “canonical neighbourhood” types, all of which turn out to be amenable to surgery.) To prove this, we argue by contradiction, assuming we can find a Ricci flow in which P fails on a sequence of points in spacetime that approach the singularity, and on some sequence of scales going to zero. We then rescale the flow to create a sequence of rescaled Ricci flows as discussed above, each of which exhibits failure of P at unit scales near the origin (here we use the hypothesis that P is scale-invariant). Now, we use compactness to find a subsequence of flows converging to an asymptotic profile . If the convergence is strong enough, the asymptotic profile will also exhibit failure of P. But now one simply goes through the list of all possible profiles in one’s classification and verifies that each of them obeys P; and one is done.

Unfortunately, just knowing that a Ricci flow is ancient and has everywhere non-negative curvature does not seem enough, by itself, to obtain a full classification of asymptotic profiles (though one can definitely say some non-trivial statements about ancient Ricci flows with non-negative curvature, most notably the Li-Yau-Hamilton inequality, which we will discuss later). To proceed further, one needs further control on asymptotic profiles . The only reasonable way to obtain such control is to obtain control on the rescaled flows which is uniform in n. While some control of this sort can be established merely by choosing the points and scales in a clever manner, there is a limit as to what one can accomplish just by point-picking alone (especially if one is interested in establishing properties P that apply to quite general regions of spacetime and general scales, rather than specific, hand-picked regions and scales). To really get good control on the rescaled flows , one needs to obtain control on the original flow which does not deteriorate when one passes from the original flow to the rescaled flow.

One can express what “does not deteriorate” means more precisely using the language of dimensional analysis, or more precisely using the concepts of subcriticality, criticality, and supercriticality from nonlinear PDE. Suppose we have some (non-negative) scalar quantity that measures some aspect of a flow . [Dimensional analysis becomes trickier when considering tensor-valued quantities, though in practice one can use the magnitude of such quantities as a scalar-valued proxy for these tensor-valued objects; see my paper on Perelman's argument for some further discussion.] In many situations, this quantity has some specific dimension k, in the sense that one has a scaling relationship

(3)

that measures how that quantity changes under the rescaling (1). In dimensional analysis language, (3) asserts that F has the units .

Assuming that F is also invariant under time translation (and under changes of spatial origin), (3) implies that

. (4)

Thus, if F is critical or dimensionless (which means that k=0) or subcritical (which means that ), any upper bound on F for the original Ricci flow will imply uniform bounds on the rescaled flows , and thus (assuming the convergence is strong enough, and F has some good continuity properties) on the asymptotic profile . In the subcritical case, F should in fact now vanish in the limit. On the other hand, if F is supercritical (which means that ) then no information about the asymptotic profile is obtained.

In order for control of to be truly useful, we would like the quantity F to be coercive. This term is not precisely defined (though it is somewhat analogous to the notion of a proper map), but coercivity basically means that upper bounds on translate to some upper bounds on various norms or similar quantities measuring the “size” of , and (hopefully) to then obtain useful bounds on the topology and geometry of .

Let us give some examples of various such quantities F for Ricci flow. We begin with some supercritical quantities:

1. Any length-type quantity, e.g. the diameter of the manifold, or the injectivity radius, has dimension 1 and is thus supercritical.
2. The various widths of 3-dimensional Ricci flows from the previous lectures, which were based on areas of minimal surfaces, have dimension 2 and are also supercritical. Thus the various bounds we have on these quantities from Lectures 4, 5, 6 do not directly tell us anything about asymptotic profiles.
3. The volume of 3-manifolds has dimension 3 and is thus also supercritical. Thus upper bounds on volume, such as Corollary 2 from Lecture 3, do not directly tell us anything about asymptotic profiles (though they are useful for other tasks, most notably for ensuring that surgery times are discrete, see Theorem 3 from Lecture 2).

As for subcritical quantities, one notable one is the minimal scalar curvature . One can check (cf. the dimensional analysis at the end of Lecture 0)
that scalar curvature has dimension -2 and is thus subcritical. The quantity , that measures the maximal amount of negative scalar curvature present in a Ricci flow, is then bounded (by the maximum principle, see Proposition 2 of Lecture 3), and so by the previous discussion will vanish for asymptotic profiles; in other words, asymptotic profiles always have non-negative scalar curvature. Unfortunately, this quantity is only partially coercive; it prevents scalar curvature from becoming arbitrarily large and negative, but does not prevent scalar curvature from becoming arbitrarily large and positive. (Also, it is possible for other curvatures, such as Ricci and Riemann curvatures, to be large even while the scalar curvature is small or even zero.) So this quantity does say something non-trivial about asymptotic profiles, but is insufficient by itself to fully control such profiles.

In the next lecture we shall see that the least eigenvalue of the modified Laplace-Beltrami operator, which can be viewed as an analytic analogue of the geometric quantity related to Poincaré inequalities, also enjoys a monotonicity property (which is connected to a certain gradient flow interpretation of (modified) Ricci flow); like , the least eigenvalue has dimension -2 and is thus also subcritical, but again it is not fully coercive, as it only prevents scalar curvature from becoming too negative.

So far we have not discussed any critical quantities. (One can create some trivial examples of critical quantities, such as the dimension or topological quantities such as , but these are not obviously coercive (the topological coercivity of the latter quantity being, of course, precisely the Poincaré conjecture that we are trying to prove!).) One way to create critical quantities is to somehow combine subcritical and supercritical examples together. Here is one simple example, due to Hamilton:

Exercise 1. Show that the quantity is critical (scale-invariant) and monotone non-increasing in time under d-dimensional Ricci flow, where denotes the volume of at time t. (This quantity can be used, for instance, to show that Ricci flow admits no “breather” solutions, i.e. non-constant periodic solutions; see the discussion in Perelman’s paper. Unfortunately, as with previous examples, it is not fully coercive.)

In the next few lectures, we will see two more advanced versions of critical controlled quantities of an analytic nature, the Perelman entropy (a scale-invariant version of the minimal eigenvalue , which is to log-Sobolev inequalities as the latter quantity is to Poincaré inequalities) and the Perelman reduced volume (which measures how heat-type kernels on Ricci flows compare against heat kernels on Euclidean space). These quantities were both introduced in Perelman’s first paper. The key feature of these new critical quantities, which distinguishes them from previously known examples, is that they are now coercive: they provide a crucial scale-invariant geometric control on a flow , which is now known as -noncollapsing. This control, which describes a relationship between the supercritical quantities of length and volume and the subcritical quantities of curvature, will be discussed next.

– Length, volume, curvature, and collapsing –

Let p be a point in a d-dimensional complete Riemannian manifold (M,g) (we make no assumptions on the dimension d here). We will establish here some basic results in comparison geometry, which seeks to understand the relationship between the Riemann curvature of the manifold M, and various geometric quantities of M such as the volume of balls and the injectivity radius, especially when compared against model geometries such as the sphere and hyperbolic space. (This is only a brief introduction; see e.g. Chapters 6, 9, and 10 of Petersen’s book for a more detailed treatment.)

Of course, in the case of Euclidean space with the Euclidean metric, the Riemann curvature is identically zero, and the volume of B(p,r) is for some explicit constant depending only on dimension. For Riemannian manifolds, it is easy to see that the volume of B(p,r) is in the limit ; for more precise asymptotics, see Exercises 7 and 8 below.

One of the most effective tools to study these questions comes from normal coordinates, or more precisely from the exponential map from the tangent space to M, defined by setting to be the value of , where is the unique constant-speed geodesic with and . By the Hopf-Rinow theorem, M is complete (in the metric sense) if and only if the exponential map is defined on all of . Henceforth we will always assume M to be complete. The ball of radius in M centred at p is then the image under the exponential map of the ball of the tangent space of the same radius (using the metric , of course):

. (5)

Thus we can study the balls centred at p by using the exponential map to pull back to the tangent space and analysing the geometry there. Two radii become relevant for this approach:

1. The injectivity radius at p is the supremum of all radii r such that is injective on .
2. The conjugate radius at p is the supremum of all radii r such that is an immersion on .

In many situations, these two radii are equal, but there are cases in which the injectivity radius is smaller. In fact the injectivity radius is always less than or equal to the conjugate radius; see Exercise 4 below.

Example 1. (Sphere) Let , and let be the sphere of radius , with the metric induced from the metric of Euclidean space . Then at every point p of M, the injectivity radius and conjugate radius are both equal to , which is also the diameter of the manifold. Note also that this manifold has constant sectional curvature K.

Example 2. (Hyperbolic space) Let , and let be hyperbolic space of hyperbolic radius , with the metric induced from the metric of Minkowski space. Then at any point p in M, e.g. p = (1,0), the injectivity radius, conjugate radius, and diameter are infinite. This manifold has constant sectional curvature -K.

Example 3. (Torus) Let , and let be the d-torus which is the product of d circles of length r. Then for any point p in M, the injectivity radius is r/2 and the conjugate radius is infinite. Here the sectional curvature is of course 0 everywhere.

The metric g on M induces a pullback metric on , which by abuse of notation we shall also call g. This metric can degenerate once one passes the conjugate radius, but let us ignore this issue for the time being. On , we have the radial variable r (defined as the magnitude of a tangent vector with respect to g(p)), and the radial vector field (defined as the dual vector field to r using polar coordinates), which is smooth away from the origin.

In Euclidean space, the vector field is the gradient of r. Happily, the same fact is true for more general Riemannian manifolds:

Lemma 1. (Gauss lemma)

1. Away from the origin, we have and .
2. Away from the origin, is the gradient of r with respect to the metric g, thus .

Exercise 2. Prove Lemma 1. (Hint: part 1 follows from the geodesic flow equation . For part 2, one way to proceed is to establish the ODE

(6)

and then apply Gronwall’s inequality.

Lemma 1 gives some important relationships between the radial vector field and the Hessian (which can be viewed as the second fundamental form of the spheres centred at p):

Exercise 3. Away from the origin, obtain the deformation formula

(7)

and the Riccati-type equation

. (8)

Also, show that has as a null eigenvector.

Exercise 4. Show that the injectivity radius of a point p cannot exceed the conjugacy radius . (Hint: there are several ways to establish this. Here is one: suppose for contradiction that , thus for some small . Let be a vector of magnitude at most . Observe that the function achieves a global minimum at whenever and so has non-negative Hessian. Use this to obtain a lower bound on on , and combine this with Exercise 3 to show that the exponential map is in fact immersed on a neighbourhood of , a contradiction. Another approach is based on Klingenberg’s inequality (see Lemma 2 below), while a third approach is based on the second variation formula for the energy of a geodesic.)

Let us now impose the bound that all sectional curvatures are bounded by some on a ball , thus

(9)

for all orthonormal tangent vectors X, Y at any point in . From Example 1 we know that the exponential map can become singular past the radius , so let us also assume that

. (10)

Note that the sectional curvature bound also implies a Ricci curvature bound for all unit tangent vectors based in .

From (9) and (10) we see that on the ball