Tags » Gradient Shrinking Solitons

285G, Lecture 17: The structure of κ-solutions

Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the previous lecture, we now use this classification, combined with extensive use of compactness and contradiction arguments, as well as the comparison geometry of complete Riemannian manifolds of non-negative curvature, to understand the structure of -solutions in these dimensions, with the aim being to state and prove precise versions of Theorem 1 and Corollary 1 from… 3,669 more words


285G, Lecture 16: Classification of asymptotic gradient shrinking solitons

In the previous lecture, we showed that every -solution generated at least one asymptotic gradient shrinking soliton . This soliton is known to have the following properties: 3,426 more words


285G, Lecture 15: Geometric limits of Ricci flows, and asymptotic gradient shrinking solitons

We now begin using the theory established in the last two lectures to rigorously extract an asymptotic gradient shrinking soliton from the scaling limit of any given -solution. 3,832 more words


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 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