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It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow.

A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of the Ricci flow on a topological $2$-sphere, then the isoperimetric ratios $C_H (\rho , t)$ of parallel loops $\gamma_{\rho}$ measured with respect to the metric $g(t)$ satisfy a heat-type equation

$\frac{\partial }{\partial t} (\log C_H) = \frac{\partial^2}{\partial {\rho}^2 } (\log C_H) + \frac{\Gamma}{L} \frac{\partial}{\partial \rho} (\log C_H)+ \frac{4 \pi - C_H}{A} \bigg( \frac{A+}{A_-} + \frac{A_-}{A_+} \bigg) $,

where $\Gamma$ is an integral of the signed curvature and more definitions can be found in the textbook of Chow and Knopf.

The isoperimetric ratio $C_H$ is similar to, but distinct from the Cheeger isoperimetric constant.

1.) Is it known if the Cheeger constant $h$ also satisfies a heat-type equation for a solution of the Ricci flow on a topological $2$-sphere?

2.) For a topological $2$-sphere, is it the case that

$C_H (M) \leq h (M)?$

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1 Answer 1

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Answer to (2) is negative. For a counterexample, consider a $2$-sphere and blow up the radius. The isoperimetric ratio $C_H$ remains the same, but the Cheeger constant $h$ shrinks, essentially because $C_H$ is a dimensionless quantity which is independent of the scaling, whereas $h$ is not.

Actually in Lemma 5.85 of the textbook of Chow and Knopf, it is shown that if $(M^2, g)$ is a closed orientable Riemannian surface, then

$C_H (M) \leq 4 \pi,$

which is obviously not true for the Cheeger constant.

I did a few calculations and looks as if answer to (1) is positive.

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    $\begingroup$ About one year ago, I feel the $h(M)$ maybe monotone along Ricci flow before singularity. But I don't know how to prove it, even though on special manifold. Since I just be a beginner of Ricci flow, I give up it. But I still be interested in it. If possible, could you tell me the calculation about (1). Thanks very much. $\endgroup$
    – Enhao Lan
    Feb 15, 2023 at 12:06

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