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285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions

In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of… 3,351 more words

Math.DG

285G, Lecture 11: κ-noncollapsing via Perelman reduced volume

Having established the monotonicity of the Perelman reduced volume in the previous lecture (after first heuristically justifying this monotonicity in Lecture 9), we now show how this can be used to establish -noncollapsing of Ricci flows, thus giving a second proof of Theorem 2 from… 3,784 more words

Math.DG

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. 3,157 more words

Math.DG

285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume

We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. 3,708 more words

Math.DG

285G, Lecture 8: Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy

It is well known that the heat equation

(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… 4,259 more words

Math.CA

285G, Lecture 7: Rescaling of Ricci flows and κ-noncollapsing

We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (Theorem 4 from Lecture 2), and turn instead to the main portion of Perelman’s argument, which is to establish the global existence result for Ricci flow with surgery (Theorem 2 from… 4,860 more words

Math.AP

285G, Lecture 6: Finite time extinction of the third homotopy group, II

In this lecture we discuss Perelman’s original approach to finite time extinction of the third homotopy group (Theorem 1 from the previous lecture), which, as previously discussed, can be combined with the finite time extinction of the second homotopy group to imply finite time extinction of the entire Ricci flow with surgery for any compact simply connected Riemannian 3-manifold, i.e. 3,519 more words

Math.AP