An energy approach to the problem of uniqueness for the Ricci flow

TL;DR

This paper presents a new energy-based proof for the uniqueness of complete Ricci flow solutions with bounded curvature and extends the result to solutions with potentially unbounded curvature using a modified energy approach.

Contribution

It introduces a simplified energy method for proving Ricci flow uniqueness and generalizes the theorem to cases with unbounded curvature.

Findings

Provided a direct energy-based proof for Ricci flow uniqueness.

Extended the uniqueness result to solutions with unbounded curvature.

Developed a variation of the energy quantity for broader applicability.

Abstract

We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqueness theorem for subsolutions to a general class of mixed differential inequalities which implies an extension of Chen-Zhu's result to solutions (and initial data) of potentially unbounded curvature.