Monotonicity of the Cheeger constant under Ricci flow on spheres

TL;DR

The paper addresses whether the Cheeger constant $h(M,g)$ improves under Ricci flow on surfaces, focusing on manifolds diffeomorphic to $S^2$. It combines evolution identities for lengths and areas along parallel foliations with a viscosity solution framework for $\log h$ to handle non-smooth switching of minimizing regions. The main result is a conditional monotonicity: if $h(M,g(t)) < 4\pi/L(\beta,g(t))$ holds for the minimizing loop $\beta$, then $h$ is non-decreasing along the flow; Papasoglu’s bound helps guarantee the condition in large-area regimes. The work also provides counterexamples showing that strict monotonicity is not guaranteed, even on spheres, reinforcing the result’s sharpness and suggesting directions for understanding non-spherical geometries and higher-genus cases.

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.

Figures (1)

• Figure 1: Pair of pants with spherical caps. The caps have positive curvature, whereas the rest of the surface has negative curvature.

Theorems & Definitions (4)

• proof
• proof
• proof
• proof