Harmonic mean curvature flow and geometric inequalities

In this talk, we use the harmonic mean curvature flow to prove Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space. Using the new Alexandrov-Fenchel type inequalities and inverse mean curvature flow, we prove the Alexandrov-Fenchel inequality for the total curvature in terms of the area for strictly convex hypersurfaces. This is a joint work with Ben Andrews (ANU) and Haizhong Li (Tsinghua).