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

## Tags » Kappa-solutions

#### 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 13: Li-Yau-Hamilton Harnack inequalities and κ-solutions

We now turn to the theory of *parabolic Harnack inequalities*, which control the variation over space and time of solutions to the scalar heat equation… 2,829 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