Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

Abstract

We prove that the isoperimetric profile of a convex domain $\Omega$ 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 $\Omega$. 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 $\text{Ric}\ge n\delta$ on $\Omega$, then the profile of $\Omega$ is bounded from above by the profile of the half-space ${\mathbb{H}}_{\delta }^{n+1}$ in the simply connected space form with constant sectional curvature $\delta$. As consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of $\Omega$, and for the first non-zero Neumann eigenvalue for the Laplace operator on $\Omega$.

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