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

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