On the question of diameter bounds in Ricci flow

Abstract: A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman's $\nu$ invariant. Here $n$ is the dimension. His result is sharp when Perelman's F entropy is positive. In this note, we give a direct proof that for all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and the Sobolev constants (or positive Yamabe constant). This bound seems directly computable in large time for some Ricci flows. In addition, since the result in its most general form is independent of Ricci flow, further applications may be possible. A generally sharp lower bound for the diameters is also given, which depends only on the initial metric, time and $L^\infty$ bound of the scalar curvature. These results imply that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows up.