Abstract:

A hypersurface in a Riemannian manifold is called 2-convex, if the sum of any two principal curvature is everywhere positive. The lecture describes joint work with Carlo Sinestrari and Simon Brendle respectively on two different parabolic geometric evolution equations for such hypersurfaces, mean curvature flow and 2-harmonic mean curvature flow, that can be used to deform the initial hypersurface diffeomorphically into shapes that can be completely classified. The smooth deformation via the geometric equation may develop singularities and the lecture explains how these singularities are overcome with surgery.