Remarks on Hamilton's Compactness Theorem for Ricci flow

TL;DR

This paper examines Hamilton's compactness theorem for Ricci flow, providing a counterexample to a commonly cited extension that assumes only local curvature bounds, thereby clarifying the theorem's limitations.

Contribution

It presents a counterexample challenging the widely accepted extension of Hamilton's compactness theorem for Ricci flow.

Findings

Counterexample shows the local curvature bound extension does not always hold

Clarifies the conditions needed for Hamilton's compactness theorem

Highlights limitations in current Ricci flow compactness results

Abstract

A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that converges to a complete limiting Ricci flow. A widely quoted extension of this result allows the curvature to be bounded uniformly only in a local sense. However, in this note we give a counterexample.