Vorträge in der Woche 21.10.2019 bis 27.10.2019

Dienstag, 22.10.2019: L-functions and graph zetas II

Julien Sessler

Donnerstag, 24.10.2019: A fully nonlinear flow of three-convex hypersurfaces

Stephen Lynch (Universität Tübingen)

We will discuss a geometric flow of compact embedded three-convex hyper- surfaces, where the normal speed at each point of the solution is given by a function of the second fundamental form. Three-convexity means that at each point, the sum of the smallest three principal curvatures is positive. This prop- erty is preserved by the flow if the speed is a concave function which vanishes on the boundary of the cone of three-convex second fundamental forms. For any such flow, the curvature of compact solutions becomes unbounded in finite time, and it is natural to ask whether solutions satisfy a convexity estimate, which implies that high-curvature regions are approximately convex. Convexity esti- mates are known to hold for mean-convex mean curvature flow, and for a large class of fully nonlinear flows of two-convex hypersurfaces. In the three-convex case, however, the concavity of the speed function is a serious impediment to establishing this kind of result. We will introduce here a new speed which, in a sense, strikes a balance between being concave and convex, and thus allows us to overcome this difficulty. The construction is quite general, and can be adapted to the k-convex setting.