Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

Abstract

We prove that the isoperimetric profile of a convex domain $Ω$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of $Ω$. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality we obtain sharp comparison theorems: not only can we derive an inequality which should be compared with Lévy-Gromov Inequality but we also show that if $\text{Ric}\geq nδ$ on $Ω$, then the profile of $Ω$ is bounded from above by the profile of the half-space $\mathbb{H}_δ^{n+1}$ in the simply connected space form with constant sectional curvature $δ$. As consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of $Ω$, and for the first non-zero Neumann eigenvalue for the Laplace operator on $Ω$.