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We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in R^3. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-White, we show that any level set flow starting from an embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, we obtain that the problem of evolving embedded 2-spheres via the mean curvature flow equation is well-posed within a natural class of singular solutions. Second, we use our result to remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows starting from any generic embedded surface only incur cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities.

 

This talk is based on joint work with Bruce Kleiner.