Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

TL;DR

This paper proves that the isoperimetric profile of compact surfaces under normalized Ricci flow remains uniformly Lipschitz continuous, ensuring stability and regularity of geometric properties during the flow.

Contribution

It establishes the uniform Lipschitz continuity of the isoperimetric profile squared for compact surfaces evolving under normalized Ricci flow, extending understanding of geometric stability.

Findings

Isoperimetric profile is jointly continuous with metric variations.

Under normalized Ricci flow, the squared isoperimetric profile is uniformly Lipschitz.

The isoperimetric profile itself is uniformly locally Lipschitz continuous.

Abstract

We show that the isoperimetric profile $h_{g(t)}(\xi)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(\xi)$ is uniform Lipschitz continuous and hence $h_{g(t)}(\xi)$ is uniform locally Lipschitz continuous.