Rigidity for critical points in the Levy-Gromov inequality

TL;DR

This paper investigates the critical points of the Levy-Gromov inequality in two-dimensional Riemannian manifolds, showing that such critical metrics are exclusively round spheres and projective planes, indicating a high rigidity in this setting.

Contribution

It characterizes the critical metrics for the Levy-Gromov inequality in two dimensions, demonstrating their rigidity and identifying them as only round spheres and projective planes.

Findings

Critical metrics in 2D are only round spheres and projective planes.

The criticality condition for the Levy-Gromov inequality is highly rigid in two dimensions.

The result narrows down the possible geometries satisfying the criticality condition.

Abstract

The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes.