Total mean curvatures of Riemannian hypersurfaces

TL;DR

This paper establishes a comparison formula for mean curvature integrals of Riemannian hypersurfaces, deriving geometric inequalities and characterizations of hyperbolic balls as minimizers in Cartan-Hadamard manifolds.

Contribution

It introduces a comparison formula for mean curvature integrals and applies it to derive inequalities and characterizations of convex hypersurfaces in non-positive curvature manifolds.

Findings

First mean curvature integral of nested hypersurfaces is bounded by that of the outer hypersurface.

Hyperbolic balls minimize mean curvature integrals among equal-radius balls in Cartan-Hadamard manifolds.

Monotonicity of mean curvature integrals extends under specific geometric conditions.

Abstract

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $\Gamma$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $\gamma$ nested inside $\Gamma$ cannot exceed that of $\Gamma$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $\Gamma$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $\gamma$ is parallel to $\Gamma$, or $M$ has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.