Geometric Analysis and Topology Seminar

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Dynamical Stability and Instability of Ricci-Flat Metrics

Speaker: Reto Mueller, Imperial College

Location: Warren Weaver Hall 517

Date: Friday, April 12, 2013, 11 a.m.

Synopsis:

Let M be a compact manifold. A Ricci-flat metric on M is a Riemannian metric with vanishing Ricci curvature. Ricci-flat metrics are fairly hard to construct, and their properties are of great interest. They are the critical points of the Einstein-Hilbert functional, the fixed points of Hamilton's Ricci flow and the critical points of Perelman's lambda-functional. In this talk, we are concerned with the stability properties of Ricci-flat metrics under Ricci flow. We will explain the following stability and instability results. If a Ricci-flat metric is a local maximizer of lambda, then every Ricci flow starting close to it exists for all times and converges (modulo diffeomorphisms) to a nearby Ricci-flat metric. If a Ricci-flat metric is not a local maximizer of lambda, then there exists a nontrivial ancient Ricci flow emerging from it. This is joint work with Robert Haslhofer.