Total mean curvatures of Riemannian hypersurfaces

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.