Isoperimetric comparisons via viscosity

TL;DR

This paper demonstrates that isoperimetric profile functions of Riemannian manifolds with Ricci lower bounds are viscosity super-solutions of certain nonlinear PDEs, leading to classical and new isoperimetric inequalities.

Contribution

It establishes a novel connection between isoperimetric profiles and viscosity solutions, enabling derivation of classical and new comparison results in Riemannian geometry.

Findings

Derives isoperimetric inequalities of Lévy-Gromov and Bérard-Besson-Gallot

Introduces viscosity super-solution framework for isoperimetric profiles

Provides new comparison results for Riemannian manifolds

Abstract

Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of L\'evy-Gromov and B\'erard-Besson-Gallot, as well as some new comparison results.