Title: Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

Authors: Cesar Rosales Lombardo and Vincent Bayle

Issue: Volume 54 (2005), Issue 5, 1371-1394

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 that 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 that should be compared with L\'evy-Gromov Inequality but we also show that if $\mathrm{Ric} \geq n\delta$ 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 a 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 $\Omega$.