| Abstract: |
| Given a smooth Ricci flow that becomes singular in finite time, a $k$-neck regions is a region of the manifold in which the flow is almost self-similar and almost splits $k$ Euclidean factors, down to arbitrarily small scales. Neck regions are characterized by a set of centres which can be thought of as a discrete approximation of high curvature regions. In this talk we will describe how, under a Type I curvature bound, we can effectively control the $k$-dimensional size of the set of centres, by carefully analyzing the behaviour of almost splitting maps at small scales.
As an application, we will see how such a result implies Perelman`s bounded diameter conjecture for a 3d Ricci flow exhibiting Type I singularities. In higher dimensions, it can be used to obtain the optimal $L^1$ curvature bound in higher dimensions, in joint work with Konstantinos Leskas. |
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