Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

TL;DR

This paper establishes new second order differential inequalities for the isoperimetric profile of convex bodies in Riemannian manifolds, leading to sharp comparison theorems and geometric estimates based on Ricci curvature bounds.

Contribution

It introduces a novel differential inequality approach to compare isoperimetric profiles in Riemannian convex bodies, extending classical inequalities and deriving geometric bounds.

Findings

Derived a second order differential inequality for the isoperimetric profile.

Established sharp comparison theorems relating profiles to space forms.

Obtained geometric estimates for volume, diameter, and eigenvalues of convex bodies.

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 L\'evy-Gromov Inequality but we also show that if $\text{Ric}\geq n\delta$ 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 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 $\Omega$.