Rigidity results, inverse curvature flows and alexandrov-fenchel type inequalities in the sphere

Abstract

We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo and Warner to convex C2-hypersurfaces. We apply these results to prove C1,B-convergence of inverse F-curvature flows in the sphere to an equator in Sn+1 for embedded, closed and strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove some Alexandrov-Fenchel type inequalities.