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285G, Lecture 14: Stationary points of Perelman entropy or reduced volume are gradient shrinking solitons

We continue our study of -solutions. In the previous lecture we primarily exploited the non-negative curvature of such solutions; in this lecture and the next, we primarily exploit the ancient nature of these solutions, together with the finer analysis of the two scale-invariant monotone quantities we possess (Perelman entropy and Perelman reduced volume) to obtain a important scaling limit of -solutions, the… 2,038 more words


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