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Dan Knopf: Lectures

May 28, 2021
TITLE: Ricci Solitons, Conical Singularities, and Nonuniqueness

In dimension n=3, there is a complete and well-posed theory of weak solutions of Ricci flow: existence in the form of “Ricci Flow Spacetimes” was proved by Kleiner and Lott, and uniqueness was proved by Bamler and Kleiner. I will describe joint work with Angenent in which we show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions n≥5. Specifically, we produce a discrete family of asymptotically conical gradient shrinking solitons, each of which encounters a finite-time local singularity and thereafter admits non-unique forward continuations by gradient expanding solitons. Moreover, we exhibit these evolutions as Ricci Flow Spacetimes and show that topological nonuniqueness after the first singularity time is possible for the solutions we construct.

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