Speaker：Julian Scheuer（Goethe University Frankfurt, Germany）

Time：2022-12-01, 15:30

Location：Tencent Meeting ID：216-291-595（No Password）

Abstract:

It is known since the seminal work by Huisken on the volume preserving mean curvature flow (VPMCF) from 1987, that the VPMCF for hypersurfaces of the sphere may lose convexity and hence a study of this flow has been avoided so far. The same problem arises for every nonlocal flow which preserves any of the other quermassintegrals. Without the presence of a global term, Guan and Li have constructed a flow within the sphere, which preserves the volume and converges smoothly to a geodesic sphere. However, the corresponding quermassintegral preserving versions of this flow are not understood until today, because it seems impossible to prove curvature estimates. In this talk, we introduce a quermassintegral preserving flow for hypersurfaces of the sphere, which is constrained by a global and a local term. This flow is shown to converge to a geodesic sphere and hence provides the first such example of a quermassintegral preserving curvature flow in the sphere, for which one can obtain smooth estimates. This is joint work with Esther Cabezas-Rivas.