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

It is well known that the heat equation


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