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Dynamical stability and instability of Ricci-flat metrics
If you have a question about this talk, please contact Neshan Wickramasekera.
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 prove 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.
This talk is part of the Geometric Analysis and Partial Differential Equations seminar series.
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