Almost periodic minimal energy blowup solutions have been constructed for a variety of critical equations, such as the nonlinear Schrodinger equation (NLS) and the nonlinear wave equation (NLW).  The formal definition of almost periodicity is that the orbit of the solution stays in a precompact subset of the energy space once one quotients out by the non-compact symmetries of the equation (namely, translation and dilation).   Another (more informal) way of saying this is that for every time , there exists a position and a frequency such that the solution is localised in space in the region and in frequency in the region , with the solution decaying in energy away from these regions of space and frequency.  Model examples of almost periodic solutions include traveling waves (in which N(t) is fixed, and x(t) moves at constant velocity) and self-similar solutions (in which x(t) is fixed, and N(t) blows up in finite time at some power law rate).

Intuitively, the reason almost periodic minimal energy blowup solutions ought to exist in the absence of global regularity is as follows.  It is known (for any of the equations mentioned above) that global regularity (and scattering) holds at sufficiently small energies.  Thus, if global regularity fails at high energies, there must exist a critical energy , below which solutions exist globally (and obey scattering bounds), and above which solutions can blow up.

Now consider a solution at the critical energy which blows up (actually, for technical reasons, we instead consider a sequence of solutions approaching this critical energy which come increasingly close to blowing up, but let’s ignore this for now).  We claim that this solution must be localised in both space and frequency at every time, thus giving the desired almost periodic minimal energy blowup solution.  Indeed, suppose is not localised in frequency at some time t; then one can decompose into a high frequency component and a low frequency component , both of which have strictly smaller energy than , and which are widely separated from each other in frequency space.  By hypothesis, each of and can then be extended to global solutions, which should remain widely separated in frequency (because the linear analogues of these equations are constant-coefficient and thus preserve frequency support).   Assuming that interactions between very high and very low frequencies are negligible, this implies that the superposition approximately obeys the nonlinear equation; with a suitable perturbation theory, this implies that is close to .  But then is not blowing up, a contradiction.   The situation with spatial localisation is similar, but is somewhat more complicated due to the fact that spatial support, in contrast to frequency support, is not preserved by the linear evolution, let alone the nonlinear evolution.

As mentioned before, this type of scheme has been successfully implemented on a number of equations such as NLS and NLW.  However, there are two main obstacles in establishing it for wave maps.  The first is that the wave maps equation is not a scalar equation: the unknown field takes values in a target manifold (specifically, in a hyperbolic space) rather than in a Euclidean space.  As a consequence, it is not obvious how one would perform operations such as “decompose the solution into low frequency and high frequency components”, or the inverse operation “superimpose the low frequency and high frequency components to reconstitute the solution”.  Another way of viewing the problem is that the various component fields of the solution have to obey a number of important compatibility conditions which can be disrupted by an overly simple-minded approach to decomposition or reconstitution of solutions.

The second problem is that the interaction between very high and very low frequencies for wave maps turns out to not be entirely negligible: the high frequencies do have a negligible impact on the evolution of the low frequencies, but the low frequencies can “rotate” the high frequencies by acting as a sort of magnetic field (or more precisely, a connection) for the evolution of those high frequencies.  So the combined evolution of the high and low frequencies is not well approximated by a naive superposition of the separate evolutions of these frequency components.

There are a number of ways to resolve the first problem.  One way, which has been pursued in a very recent paper by Sterbenz and Tataru (and also in an earlier paper of Tataru, and of myself in the case of spherical targets), is to embed the target manifold into Euclidean space and perform various operations (e.g. Littlewood-Paley projections) on the solution in that ambient space, thus creating new fields which lie outside the target.  This does not work directly with the hyperbolic space target because this is not efficiently embeddable into Euclidean space (although it was recently pointed out to me by Jacob Sterbenz that one can proceed – at least for the narrow question of establishing global regularity rather than scattering – by passing from hyperbolic space to a compact quotient).  Another approach, introduced by Krieger, is that of dynamic separation – to isolate a “dynamic” scalar field which is unconstrained, controls all the other components of the evolution (as “static” functions of the dynamic field), and then manipulate the dynamic field directly.  It is this latter approach which we will pursue in the sequel to this paper; the dynamic field we will use is the tension field of the harmonic map heat flow.

For the second problem, we will use a variant of the “frequency truncation method” of Bourgain, constructing the solution iteratively, in a sequence of time intervals in which the low frequency solution is small in a certain spacetime norm sense.   On each such time interval, the impact of the low frequencies on the high ones is small enough that one can basically ignore the low frequencies, and evolve the high frequencies using the hypothesis that solutions with energy less than have global solutions.  But then one has to appeal to the hypothesis again at every time interval, otherwise the cumulative effect of the low frequencies on the high frequencies will become uncontrollable.  This is not a problem unless the energy of the high frequencies increases significantly after every time interval.  But one can use the energy conservation of the low frequencies and of the entire solution to prevent this from occuring.

All of these things will be detailed in the sequel to the current paper.   What the current paper does is to perform the abstract argument that constructs minimal-energy blowup solutions, assuming four black-box results which will be proven in the sequel:

1. A perturbation theory for wave maps which enjoys a certain fungibility property, which is technical to state but roughly asserts that any large wave map on a long time interval can be subdivided into a controlled number of shorter time intervals in which the evolution behaves like the linear equation;
2. A means of synthesising solutions from frequency-delocalised data from solutions at strictly lower energies;
3. A means of synthesising solutions from spatially-dispersed data from solutions at strictly lower energies;
4. A means of synthesising solutions from spatially-delocalised data from solutions at strictly lower energies.

(Here, “spatially dispersed” means, roughly, that the energy density does not accumulate at any point, while “spatially delocalised” means that there is a location where the energy density accumulates, but a significant amount of energy is also present at a large distance from this accumulation point.  These two scenarios complement the scenario we actually want, which is spatial localisation – where the energy density accumulates at one point and decays away from that point.)

The abstract component of the argument is in fact quite similar to that used for the energy-critical NLS by Colliander, Keel, Staffilani, Takaoka, and myself.  There are also some more concrete components to the argument in this paper, though, namely the use of the harmonic map heat flow as a kind of nonlinear Littlewood-Paley resolution in order to formally define frequency delocalisation and set up its basic properties, and also a Rellich-type compactness lemma which asserts that solutions which are localised in space and frequency are indeed almost precompact, and which is also proven using the harmonic map heat flow.

To explain the inverse theorem, let me first discuss the bilinear estimate that it inverts.  Define a wave to be a solution to the free wave equation .  If the wave has a finite amount of energy, then one expects the wave to disperse as time goes to infinity; this is captured by the Strichartz estimates, which establish various spacetime bounds on such waves in terms of the energy (or related quantities, such as Sobolev norms of the initial data).  These estimates are fundamental to the local and global theory of nonlinear wave equations, as they can be used to control the effect of the nonlinearity.

In some cases (especially in low dimensions and/or low regularities, and with equations whose nonlinear terms contain derivatives), Strichartz estimates are too weak to control nonlinearities; roughly speaking, this is because waves decay too slowly in low dimensions.  (For instance, one-dimensional waves do not decay at all.)  However, it has been understood for some time that if the nonlinearity has a special null structure, which roughly means that it consists only of interactions between transverse waves rather than parallel waves, then there is more decay that one can exploit.  For instance, while one-dimensional waves do not decay in time, the product between a left-propagating wave and a right-propagating wave does decay in time.  In particular, if f and g are bounded in , then this product is bounded in spacetime , thanks to the Fubini-Tonelli theorem.

There is a similar “bilinear ” estimate for products of transverse waves in higher dimensions.  This estimate is the basic building block for the bilinear estimates and their variants as developed by Bourgain, Klainerman-Machedon, Kenig-Ponce-Vega, Tataru, and others, and which are the tool of choice for establishing local and global control on nonlinear wave equations, particularly at low dimensions and at critical regularities.  In particular, these estimates (or more precisely, a complicated variant of these estimates in sophisticated function spaces, due to Tataru and myself), are used in the theory of the energy-critical wave map equation.  [These bilinear (and trilinear) estimates are not, by themselves, enough to handle this equation; one also needs an additional gauge fixing procedure before the equation is sufficiently close to linear in behaviour that these estimates become effective.  But I do not wish to discuss the (significant) gauge fixing issue here.]

To cut a (very) long story short, these estimates, when combined with a suitable perturbative theory, allow one to control energy-critical wave maps as long as the energy is small.  However, the whole point of the “heatwave” project is to control the non-perturbative setting when the energy is large (but finite), and one wants to control the solution for long periods of time.

In my previous “heatwave” paper, in which I established large data local well-posedness for this equation, I finessed this issue by localising time to very short intervals, which made certain spacetime norms small enough for the perturbation theory to apply.  This sufficed for the local well-posedness theory,  but is not good enough for the global perturbative theory, because the number of very short intervals needed to cover the entire time axis becomes unbounded.  For that, one needs the ability to make certain norms or estimates “small” by only chopping up time into a bounded number of intervals.  I refer to this property as divisibility (I used to refer to it, somewhat incorrectly, as fungibility).

In the case of semilinear wave (or Schrödinger equations), in which Strichartz estimates are already sufficient to obtain a satisfactory perturbative theory, divisibility is well-understood, and boils down to the following simple observation: if a function obeys a global spacetime integrability bound such as

for some finite exponent p and some finite bound M, then one can partition into intervals I on which

for some at one’s disposal to select.  Indeed the number of such intervals is bounded by , and the intervals can be selected by a simple “greedy algorithm” argument.  This divisibility property of -type spacetime norms allows one to easily generalise the small-data perturbation theory to the large-data setting, and is relied upon heavily in the modern theory of the critical nonlinear wave and Schrödinger equations; see for instance this survey of Killip and Visan.

Unfortunately, the function spaces used in wave maps are not easily divisible in this manner (very roughly speaking, this is because the function space norms contain too many type norms within them).   So one cannot rely purely on refining the function space; one must also work on refining the bilinear (and trilinear) estimates on these spaces.   The standard way to do this is to strengthen the exponents in these estimates, and for the basic bilinear estimate this has indeed been done (in work of Wolff and myself).  This suffices for “equal-frequency” interactions, in which one is multiplying two transverse waves of the same frequency, but turns out to be inadequate for “imbalanced-frequency” interactions, when one is multiplying a low-frequency wave by a high-frequency transverse wave.  For this, I rely instead on establishing an inverse theorem for the estimate.

Generally speaking, whenever one is faced with an estimate, e.g. a linear estimate

one can pose the inverse problem of trying to classify the functions f for which the estimate is tight in the sense that

for some which is not too small.  Such inverse theorems are a current area of study in additive combinatorics, and have recently begun making an appearance in PDE as well.  For instance:

• Young’s inequality or the Hausdorff-Young inequality , is only tight (for non-endpoint p,q,r) when f, g are concentrated on balls, arithmetic progressions, or Bohr sets (this is a consequence of several basic theorems in additive combinatorics, including Freiman’s theorem and the Balog-Szemeredi-Gowers theorem);
• The trivial inequality for the Gowers uniformity norms is only expected to be tight when f correlates with a highly algebraic object, such as a polynomial phase or nilsequence (this is the inverse conjecture for the Gowers norm, which is partially proven so far);
• The Sobolev embedding is only tight when f is concentrated on a unit ball (for non-endpoint estimates) or a ball of arbitrary radius (for endpoint estimates);
• Strichartz estimates are only tight when f is concentrated on a ball (for non-endpoint estimates) or a tube (for endpoint estimates).

Inverse theorems for such estimates as Sobolev inequalities and Strichartz estimates are also closely related to the theory of concentration compactness and profile decompositions; see this previous blog post of mine for a discussion.

I can now state informally, the main result of this paper:

Theorem 1 (informal statement).  A bilinear estimate between two waves of different frequency is only tight when the waves are concentrated on a small number of light rays.  Outside of these rays, the norm is small.

This leads to a corollary which will be used in my final heatwave paper:

Corollary 2 (informal statement).  Any large-energy wave can have its time axis subdivided into a bounded number of intervals, such that on each interval the bilinear estimates for that wave (when interacted against any high-frequency transverse wave) behave “as if” was small-energy rather than large energy.

The method of proof relies on a paper of mine from several years ago on bilinear estimates for the wave equation, which in turn is based on a celebrated paper of Wolff.   Roughly speaking, the idea is to use wave packet decompositions and the combinatorics of light rays to isolate the regions of spacetime where the waves are concentrating, cover these regions by tubular neighbourhoods of light rays, then remove the light rays to reduce the energy (or mass) of the solution and iterate.  The wave packet analysis is moderately complicated, but fortunately I can use a proposition on this topic from my paper as a black box, leaving only the other components of the argument to write out in detail.

This local result is closely related to the small energy global regularity theory developed in recent years by myself, by Krieger, and by Tataru.  In particular, the complicated function spaces used in that paper (which ultimately originate from a precursor paper of Tataru).  The main new difficulties here are to extend the small energy theory to large energy (by localising time suitably), and to establish continuous dependence on the data (i.e. two solutions which are initially close in the energy topology, need to stay close in that topology).  The former difficulty is in principle manageable by exploiting finite speed of propagation (exploiting the fact (arising from the monotone convergence theorem) that large energy data becomes small energy data at sufficiently small spatial scales), but for technical reasons (having to do with my choice of gauge) I was not able to do this and had to deal with the large energy case directly (and in any case, a genuinely large energy theory is going to be needed to construct the minimal energy blowup solution in the next paper).  The latter difficulty is in principle resolvable by adapting the existence theory to differences of solutions, rather than to individual solutions, but the nonlinear choice of gauge adds a rather tedious amount of complexity to the task of making this rigorous.  (It may be that simpler gauges, such as the Coulomb gauge, may be usable here, at least in the case of the hyperbolic plane (cf. the work of Krieger), but such gauges cause additional analytic problems as they do not renormalise the nonlinearity as strongly as the caloric gauge.  The paper of Tataru establishes these goals, but assumes an isometric embedding of the target manifold into a Euclidean space, which is unfortunately not available for hyperbolic space targets.)

The main technical difficulty that had to be overcome in the paper was that there were two different time variables t, s (one for the wave maps equation and one for the heat flow), and three types of PDE (hyperbolic, parabolic, and ODE) that one has to solve forward in t, forward in s, and backwards in s respectively.  In order to close the argument in the large energy case, this necessitated a rather complicated iteration-type scheme, in which one solved for the caloric gauge, established parabolic regularity estimates for that gauge, propagated a “wave-tension field” by the heat flow, and then solved a wave maps type equation using that field as a forcing term.  The argument can eventually be closed using mostly “off-the-shelf” function space estimates from previous papers, but is remarkably lengthy, especially when analysing differences of two solutions.  (One drawback of using off-the-shelf estimates, though, is that one does not get particularly good control of the solution over extended periods of time; in particular, the spaces used here cannot detect the decay of the solution over extended periods of time (unlike, say, Strichartz spaces for ) and so will not be able to supply the long-time perturbation theory that will be needed in the next paper in this series.  I believe I know how to re-engineer these spaces to achieve this, though, and the details should follow in the forthcoming paper.)

1. A construction of an energy space for maps into hyperbolic space obeying a certain set of reasonable properties, such as compatibility with symmetries, approximability by smooth maps, and existence of a well-defined stress-energy tensor.
2. A large data local well-posedness result for wave maps in the above energy space.
3. The existence of an almost periodic “minimal-energy blowup solution” to the wave maps equation in the energy class, if this equation is such that singularities can form in finite time.
4. The non-existence of any non-trivial degenerate maps into hyperbolic space in the energy class, where “degenerate” means that one of the partial derivatives of this map vanishes identically.
5. The non-existence of any travelling or self-similar solution to the wave maps equation in the energy class.

In this paper, the second of four in this series (or, as the title suggests, the fourth in a series of six papers on wave maps, the first two of which can be found here and here), I verify Claims 1, 4, and 5.  (The third paper in the series will tackle Claim 2, while the fourth paper will tackle Claim 3.)  These claims are largely “elliptic” in nature (as opposed to the “hyperbolic” Claims 2, 3), but I will establish them by a “parabolic” method, relying very heavily on the harmonic map heat flow, and on the closely associated caloric gauge introduced in an earlier paper of mine.  The results of paper can be viewed as nonlinear analogues of standard facts about the linear energy space , for instance the fact that smooth compactly supported functions are dense in that space, and that this space contains no non-trivial harmonic functions, or functions which are constant in one of the two spatial directions.  The paper turned out a little longer than I had expected (77 pages) due to some surprisingly subtle technicalities, especially when excluding self-similar wave maps.  On the other hand, the heat flow and caloric gauge machinery developed here will be reused in the last two papers in this series, hopefully keeping their length to under 100 pages as well.

A key stumbling block here, related to the critical (scale-invariant) nature of the energy space (or to the failure of the endpoint Sobolev embedding ) is that changing coordinates in hyperbolic space can be a non-uniformly-continuous operation in the energy space.  Thus, for the purposes of making quantitative estimates in that space, it is preferable to work as covariantly (or co-ordinate free) manner as possible, or if one is to use co-ordinates, to pick them in some canonical manner which is optimally adapted to the tasks at hand.  Ideally, one would work with directly with maps (as well as their velocity field ) without using any coordinates on , but then it becomes to perform basic analytical operations on such maps, such as taking the Fourier transform, or (even more elementarily) taking the difference of two maps in order to measure how distinct they are from each other.

Fortunately, the harmonic map heat flow can resolve a lot of these problems.  Thanks to the negative curvature of the target manifold , one can show that any (finite energy) map will contract under harmonic map heat flow to a single point (or more precisely, to a constant map).  This result (essentially due to Eells and Sampson) is consistent with the fact that does not support any non-trivial finite energy harmonic maps (in contrast with positive curvature targets, such as the sphere), and is ultimately derived from a version of the Bochner-Weitzenböck identity.

When the map has become constant, one can put a constant orthonormal frame on it.  Running the heat flow backwards in time, and dragging back this frame, one obtains a canonical frame (up to a rotation of the entire frame) to place on the original map, which is remarkably “flat”, in the sense that its connection coefficients are small in various function space norms.  I call this frame the “caloric gauge” for the map (as opposed to other frames one can place on such maps, such as the Coulomb gauge, radial gauge, Lorenz gauge, or Cronstrom gauge).  It has the advantage of being well-defined and essentially unique even for large energy maps, so long as the target is negatively curved. (The situation here is analogous to that of using Ricci flow to understand the geometry of manifolds, as in Perelman’s proof of the Poincaré conjecture, though things are enormously simpler in the setting of harmonic map heat flows into hyperbolic space due to the lack of topological obstructions (the domain is contractible) or geometric obstructions (no non-trivial harmonic maps, which are the analogue of Ricci solitons); also, the heat flow equation is semilinear rather than quasilinear, with the geometry of the background domain being fixed rather than evolving).

When viewed in this gauge, the heat flow resembles a nonlinear version of the linear heat equation, and the velocity field (which now takes values in the vector space ) can be viewed as a nonlinear Littlewood-Paley resolution of the original map.  It then becomes possible to define the energy space (and its attendant metric structure) using this resolution in exact analogy with standard Littlewood-Paley theory.  One can then verify Claim 1 by a heavy use of parabolic regularity estimates.

Claims 4 and 5 are proven by the strategy of first applying the heat flow for a short amount of time to regularise the solution to the extent that formal computations can be justified rigorously (with all error terms incurred being manageable), and then adapting whatever arguments work in the smooth case to this regularised energy space setting.  For instance, a non-trivial smooth finite energy map cannot have vanishing derivative in some direction, as this would cause the map to be constant for arbitrarily amounts of displacement in this direction, leading to an infinite amount of energy in the map.  This argument can be adapted to regularised energy class maps, using a one-dimensional Poincaré inequality in that direction.

To rule out travelling wave maps (the first part of Claim 5), the idea is to represent each such travelling wave map as a (Lorentz contracted) harmonic map, and then use standard arguments (based on the Bochner-Weitzenböck identity) to show such maps are trivial.  [In principle, one could use Lorentz transforms to send the velocity of the wave map to zero, but I had difficulty making these transforms cooperate with the initial value problem or the caloric gauge, and eventually abandoned any use of these transformations.] To do this one must first regularise the wave map via heat flow in order to be able to use the wave map equation classically (rather than in some weak or distributional sense), as this is necessary in order to establish the harmonic map property.  Unfortunately, the heat flow and wave map equation do not quite commute, and so one has to measure this failure of commutativity quite precisely (and in particular, obtain bounds and uniform continuity for second time derivatives of wave maps under heat flow, which turns out to be rather delicate technically), and one eventually ends up with an approximate harmonic map rather than an exaclty harmonic map.  Fortunately, one can use the heat flow again to show that such maps are nearly trivial, which suffices to establish the first part of Claim 5.

Similarly, by using hyperbolic polar coordinates one sees that self-similar wave maps are (formally) equivalent to harmonic maps on hyperbolic space, which can then be converted to a harmonic map on the unit disk by a conformal transformation.  A classical argument of Lemaire shows that such maps are trivial if they are smooth and vanish on the boundary.  I was unable to use this argument directly – it requires too much regularity (in particular, exploiting unique continuation, which is very expensive in regularity and not particularly stable) – but a heat flow argument turns out to work well instead.  In order to keep boundary effects manageable, though, it was necessary to obtain additional decay of angular derivatives near this boundary, which we achieved using the holomorphicity of the Hopf differential (which was also a key component of Lemaire’s argument).

Global regularity of wave maps III. Large energy from to hyperbolic spaces

For the last nine years or so, I have been working on and off on the global regularity problem for wave maps . The wave map equation is a nonlinear generalisation of the wave equation in which the unknown field takes values in a Riemannian manifold rather than in a vector space (much as the concept of a harmonic map is a nonlinear generalisation of a harmonic function). This equation (also known as the nonlinear model) is one of the simplest examples of a geometric nonlinear wave equation, and is also arises as a simplified model of the Einstein equations (after making a U(1) symmetry assumption). The global regularity problem seeks to determine when smooth initial data for a wave map (i.e. an initial position and an initial velocity tangent to the position) necessarily leads to a smooth global solution.

The problem is particularly interesting in the energy-critical dimension d=2, in which the conserved energy becomes invariant under the scaling symmetry . (In the subcritical dimension d=1, global regularity is fairly easy to establish, and was first done by Gu and by Ladyzhenskaya-Shubov; in supercritical dimensions , examples of singularity formation are known, starting with the self-similar examples of Shatah.)

It is generally believed that in two dimensions, singularities can form when M is positively curved but that global regularity should persist when M is negatively curved, in analogy with known results (in particular, the landmark paper of Eells and Sampson) for the harmonic map heat flow (a parabolic cousin of the wave map equation). In particular, one should always have global regularity when the target is a hyperbolic space. There has been a large number of results supporting this conjecture; for instance, when the target is the sphere, examples of singularity formation have recently been constructed by Rodnianski-Sterbenz and by Krieger-Schlag-Tataru, while for suitably negatively curved manifolds such as hyperbolic space, global regularity was established assuming equivariant symmetry by Shatah and Tahvildar-Zadeh, and assuming spherical symmetry by Christodoulou and Tahvildar-Zadeh. I will not attempt to mention all the other results on this problem here, but see for instance one of these survey articles or books for further discussion.

Back in 2001, I managed to show that one has global regularity for this problem when the target is a sphere and the energy was sufficiently small (building on an earlier result of mine in higher dimensions, and on a Besov space variant of the result by Tataru). The main new innovation here was a “microlocal gauge transform” that made the equation slightly less nonlinear, enough so that perturbative techniques become effective. (This work was recognised with the 2002 Bôcher Prize.) A simpler and more geometric gauge transform (the Coulomb gauge) was then introduced by Shatah-Struwe and by Nahmod-Stefanov-Uhlenbeck, and the results extended to a wide variety of other manifolds by these authors (and by Klainerman-Rodnianski, Krieger, and Tataru).

In recent years, there have been a number of methods that can extend small energy regularity results to large energy in the case when the energy is scale-invariant. For instance, for the energy-critical wave equation, an energy non-concentration argument based primarily on Morawetz-type inequalities (which in turn arise from an analysis of the stress-energy tensor), combined with the local (or small energy) theory (based primarily on Strichartz estimates), was able to handle the large-energy case, as worked out some time ago by Grillakis, Shatah-Struwe, and others. These methods could handle the wave maps equation under additional symmetry assumptions, but the available Morawetz inequalities appeared to be giving the “wrong” sort of control on the solution to handle the general case (just as control of a function in one function space norm does not automatically imply control in other norms).

However, in 1999, Bourgain introduced a new tool (the induction on energy method) which allowed one to convert one type of control on a solution to another to obtain global regularity results for critical PDE. This method was clarified by a number of subsequent papers, including one by Colliander, Keel, Staffilani, Takaoka, and myself, and one by Kenig and Merle, as identifying the “minimal energy blowup solution” for any given PDE for which singularities can develop, using large data perturbation theory to show that such a solution is necessarily almost periodic (modulo the symmetries of the equation), and then using global methods such as Morawetz estimates to rule out the existence of such solutions.

These new tools were applied to scalar models such as NLS or NLW and were not immediately applicable to the wave maps equation. Nevertheless, a potential strategy to the large data global regularity wave map became visible: firstly, one had to extend my small energy regularity theory to a large energy perturbation theory; secondly, one had to locate a Morawetz-type estimate to control minimal energy blowup solutions; and thirdly one had to adapt the induction-on-energy method to the wave map setting.

By 2004, I had managed to locate a promising new gauge to renormalise the wave maps equation based on the harmonic map heat flow, which I called the “caloric gauge”, and which (in contrast to previous gauges) was able to handle large energy solutions in the case of negatively curved targets thanks to the work of Eells and Sampson mentioned earlier. In principle, by splicing this gauge into my earlier paper I would be able to obtain the large energy perturbation theory needed for the above program. In my 2004 paper, I was also able to find a Morawetz estimate (extending some earlier such estimates in the symmetric setting; a similar estimate had also been obtained by Grillakis) that suggested that wave maps became asymptotically self-similar as one approached any given singularity. Since it had been established by Shatah and Struwe that no genuinely self-similar wave maps existed, this in principle provided the second ingredient in the program.

Also about this time, I managed to convince myself that the induction on energy argument could be adapted to the wave map setting. A major new complication here is that the unknown fields are no longer scalar fields, but instead take values in a manifold (or, if one takes derivatives, they become vector fields but then need to obey a number of additional constraint equations that make the system rather rigid). Because of this, some of the fundamental tools in the induction on energy strategy, such as the use of cutoffs in space or frequency to decompose a field into localised components, as well as the use of the humble addition operation to superimpose such components back together to reconstitute the solution, had to be reworked from scratch. By appealing again to the harmonic map heat flow, I was able to find substitutes for all of these operations, which in principle gave the third ingredient in the program.

However, it was clear that putting all of this together would be an enormous task. For comparison, my small-energy paper (the prototype for the first step in the program) was 102 pages long, and my paper with Colliander et al. (my initial prototype for the third step) was 100 pages long. (The second step was not nearly as fearsome, though still nontrivial: my caloric gauge paper was 32 pages long, and the argument that rules out self-similar wave maps can be written in a handful of pages.) To make matters worse, the task of combining the arguments was more multiplicative in nature than additive, as the basic building blocks of each argument had to be reworked to accomodate the complications of the other. I was beginning to estimate the total length of the paper to run at perhaps 500 pages. I spent some months writing nearly a hundred pages of (unpublished) notes towards this goal, but eventually got exhausted (as well as distracted by many other things) and essentially shelved the project for several years (though I did once get a remarkable offer to run a workshop specifically designed to finally execute the various components of this program!).

As it turns out, though, this procrastination was the right thing to do, because several new conceptual advances and simplifications in the field occurred in the meantime. For instance, the 2006 paper of Kenig and Merle mentioned earlier managed to eliminate a lot of tedious “epsilon management” from the arguments of Colliander et al. (though for a slightly different problem), by using dispersive analogues of the theory of concentration compactness, and in particular the use of linear profile decompositions. When I started working with Killip, Visan, and Zhang on applying the Kenig-Merle methods to the mass-critical NLS, we realised that the method allowed for the global regularity problem to be cleanly “factored” into two non-interacting components: a reduction to almost periodic solutions using the large data perturbation theory, and a Morawetz inequality-based argument ruling out the existence of non-trivial almost periodic solutions. (This philosophy had already been adopted for some time by Merle and his co-authors for some slightly different dispersive models.) From this, I estimated that my previously planned 500 page paper could now be replaced with two papers of 100-200 pages in length each – still not exactly a pleasant prospect, but a significant amount of progress nevertheless, especially given that I had done very little direct work on the project for some years.

Still, the amount of work required was sufficiently daunting that I continued to postpone the actual writing process, in favour of shorter projects that offered a more immediate payoff. This quarter, however, as I was teaching my class on Perelman’s proof of the Poincaré conjecture, I realised that Perelman’s three-tier approach of understanding singularities of Ricci flow – by passing from general Ricci flows to ancient -solutions and then to asymptotic gradient shrinking solitons – could be adapted to the problem of studying almost periodic wave maps, passing from such wave maps to ancient wave maps and then to self-similar, stationary, or travelling wave maps, and reducing matters to ruling out the existence of the latter type of wave map in the energy class. Furthermore, the argument was rather abstract: the large data perturbation theory that I needed for it could be encapsulated into a cleanly formulated hypothesis (which was highly plausible based on known results of this type for other dispersive models). This removed a psychological block from my task of writing down the whole argument, because the large data perturbation theory is one of the lengthiest components of the proof (being based on my 102 page small-energy paper, which unfortunately has not really been simplified too much despite a number of generalisations and refinements) and had discouraged me from even beginning the process.

Accordingly, in the paper I have now uploaded to the arXiv (a “mere” 35 pages long), I have written down the “high-level” component of the argument, which shows how global regularity for large energy wave maps in the model case when the target manifold is hyperbolic space follows from five simpler claims, which roughly speaking are as follows:

1. A construction of a suitable energy space (a nonlinear analogue of the Sobolev space ) with some reasonable properties;
2. A large data local well-posedness result in this energy space;
3. The conclusion of the induction-on-energy argument, namely that lack of global regularity implies existence of a non-trivial almost periodic solution;
4. The non-existence of self-similar, stationary, or travelling wave maps in the energy class; and
5. The non-existence of a energy class function which splits into the tensor product of a function of one lower dimension and a constant.

With these claims, and repeated use of compactness arguments (in the spirit of Perelman) and the conservation of the stress-energy tensor (in particular using the Morawetz estimate from my caloric gauge paper to create asymptotic self-similarity of the ancient solution), the current paper establishes the global regularity for wave maps into hyperbolic space. The basic strategy, once one has the above five ingredients, follows the Perelman approach (but is greatly simplified by the lack of singularities, which of course are the major obstacle in understanding Ricci flow):

• Assume for contradiction that global regularity fails; then (by 3.) there is a non-trivial almost periodic solution.
• By rescaling this solution and taking limits (using 2.), one can extract an ancient almost periodic solution.
• By using the Morawetz estimate, show that this ancient solution becomes asymptotically self similar as one moves backwards in time.
• Rescaling and taking limits again, obtain either a self-similar solution, a stationary solution, or a travelling solution, which travels either below or at the speed of light.
• The first three cases can be eliminated by 4. In the last case of a solution travelling at the speed of light, it turns out that Lorentz contraction forces the solution to split into a solution of one lower dimension and a constant, which can then be eliminated by 5.

In the near future I plan to complete three more papers on this topic, devoted to the claims 1-5 above; I estimate each of the papers to be 30-100 pages long. Hopefully, by dividing the project up into more manageable chunks, it should be completed at a much faster rate than previously.