It was shown by [Hamilton](https://www.degruyter.com/document/doi/10.1515/9781400882571-013/html#:%7E:text=AN%20ISOPERIMETRIC%20ESTIMATE%20FORTHE%20RICCI%20FLOWON%20THE%20TWO-SPHERE,%E2%80%94%20RQij%20where%20R%20is%20the%20scalar%20curvature.) 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](https://en.wikipedia.org/wiki/Cheeger_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


$C\_H (M) \leq h (M)?$



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


