Monotonicity of the Cheeger constant under Ricci flow on spheres

TL;DR

This paper investigates how the Cheeger isoperimetric constant behaves under Ricci flow on spheres, proving it is non-decreasing and providing examples where it remains constant, highlighting nuanced behavior.

Contribution

It establishes the non-decreasing nature of the Cheeger constant under Ricci flow on spheres and introduces a viscosity approach to handle region switching.

Findings

Cheeger constant is non-decreasing along Ricci flow on spheres

Examples show the Cheeger constant can remain constant during the flow

Viscosity formulation helps analyze the evolution of the Cheeger constant

Abstract

We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to $S^2$, we show that the Cheeger constant is non-decreasing along the flow. The proof uses evolution identities for parallel curves together with a viscosity formulation of the evolution of $\log h$ which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological $2$-spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected.