On the question of diameter bounds in Ricci flow

TL;DR

This paper establishes diameter bounds for compact manifolds under Ricci flow based on scalar curvature bounds, volume, and Sobolev constants, providing a direct and computable approach that applies generally beyond Ricci flow.

Contribution

It offers a new direct proof that diameter bounds depend only on scalar curvature bounds, volume, and Sobolev constants, independent of previous entropy-based conditions.

Findings

Diameter bounds depend on scalar curvature, volume, and Sobolev constants.

Finite-time diameter cannot become infinite or zero unless scalar curvature blows up.

Provides a sharp lower bound for diameter based on initial metric and scalar curvature.

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.