The Ricci flow and isoperimetric inequalities on surfaces

We revisit the connection between the Ricci flow and isoperimetric inequalities on surfaces which are diffeomorphic to the $2$-sphere. We prove that the Cheeger isoperimetric constant is non-decreasing under Ricci flow on topological $2$-spheres. A topological $2$-sphere with non-trivial curvature is exhibited which is a counterexample to the hypothesis that the Cheeger constant is a strictly increasing function of the Ricci flow.