Harmonic mean curvature flow and geometric inequalities

Abstract

We employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total curvature, which is the integral of Gaussian curvature on the hypersurface. The resulting inequality allows us to use the inverse mean curvature flow to prove Alexandrov-Fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.