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\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.

Keywords

• reilly’s formulas
• quermassintegral
• mixed volume
• generalized mean curvature
• hyperbolic space
• cartan-hadamard manifold
• primary: 53c20
• 58j05
• secondary: 52a38
• 49q15