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 $Γ$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $γ$ nested inside $Γ$ cannot exceed that of $Γ$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $Γ$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $γ$ is parallel to $Γ$, 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.