  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.
