Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere

TL;DR

This paper establishes a rigidity result in the sphere, extends convex hypersurface results to less smooth cases, analyzes inverse curvature flows, and proves Alexandrov-Fenchel type inequalities in spherical geometry.

Contribution

It generalizes a convex hypersurface rigidity result to $C^2$-hypersurfaces and applies inverse curvature flows to derive geometric inequalities in the sphere.

Findings

Convergence of inverse F-curvature flows to an equator in the sphere.

Extension of rigidity results to $C^2$-hypersurfaces.

Proof of Alexandrov-Fenchel type inequalities in spherical geometry.

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-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,\beta}$-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedded, closed, 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 Alexandrov-Fenchel type inequalities in the sphere.