A lower bound for the diameter of solutions to the Ricci flow with nonzero $H^{1}(M^{n};R)$

TL;DR

This paper establishes a lower bound on the diameter of Ricci flow solutions on compact manifolds with nonzero first cohomology, confirming Hamilton's conjecture about product metrics on $S^{1} imes S^{n-1}$.

Contribution

It provides the first lower bound for the diameter of Ricci flow solutions under the condition of nonvanishing first cohomology, and confirms a longstanding conjecture.

Findings

Lower bound for diameter of Ricci flow solutions established.

Hamilton's conjecture on product metrics confirmed.

Implications for the geometry of Ricci flow limits.

Abstract

We obtain a lower bound for the diameter of a solution to the Ricci flow on a compact manifold with nonvanishing first real cohomology. A consequence of our result is an affirmative answer to Hamilton's conjecture that a product metric on $(S^{1}\times S^{n-1}$ cannot arise as a final time limit flow.