# Does the Cheeger constant satisfy a heat-type equation?

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

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

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
$$C_H (M) \leq 4 \pi,$$
• 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. Feb 15, 2023 at 12:06