Geometry Seminar: P. Gianniotis (National and Kapodistrian University of Athens)
Geometry Seminar (Geometric Analysis)
Speaker: Panagiotis Gianniotis (National and Kapodistrian University of Athens)
Title: Neck regions and Perelman's bounded diameter conjecture for Type I Ricci flows
Speaker: Panagiotis Gianniotis (National and Kapodistrian University of Athens)
Title: Neck regions and Perelman's bounded diameter conjecture for Type I Ricci flows
Abstract: In this talk we discuss some new results on the size of k-neck regions in a Ricci flow, under Type I bounds on the curvature. There are regions where the flow is approximately selfsimilar down to arbitrarily small scales, approximately splitting exactly k Euclidean factors, and are characterized by a set of centers which can be thought of as a discrete approximation of the singular set. It turns out that, by carefully analyzing the behaviour of almost splitting maps at small scales, we can effectively control the k-dimensional size of the set of centers.
As an application, we describe how our results can be used to confirm Perelman’s bounded diameter conjecture for a 3d Ricci flow exhibiting Type I singularities. Moreover, we will also discuss joint work with Konstantinos Leskas, where we use these results to show that, under a Type I bound on the curvature, the curvature tensor is uniformly bounded in L^1, in any dimension.