Paper · Apr 12, 2024

Monotonicity of the Cheeger constant under Ricci flow on spheres

Hollis Williams

math.DGarXiv:2404.13063v2
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Abstract

We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to , 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 which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological -spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected.

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Submitted
Apr 12, 2024
Last revised
Nov 14, 2025 (v2)
Subjects
math.DG

Comments: More careful proof of main theorem added

Ricci flowSurfacesGeometric inequalities

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  7. Notes on harmonic-Ricci flow on surface

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    Topics Surfaces

  8. Ricci flow on surfaces along the standard lightcone in the -Minkowski spacetime

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.