Hamilton's identity and rigidity of complete gradient solitons

TL;DR

This paper extends Hamilton's identity to general geometric flows, providing new insights into the rigidity, potential function growth, and volume properties of gradient solitons beyond Ricci flows.

Contribution

It introduces a generalized Hamilton's identity applicable to various geometric flows, enabling the extension of known Ricci soliton results to broader contexts.

Findings

Hamilton's identity is applicable to general geometric flows.

Results on rigidity and volume growth are extended beyond Ricci solitons.

The approach recovers known properties like the Omori-Yau maximum principle for new flows.

Abstract

In this work, we study gradient solitons to general geometric flows. Our approach is to understand what assumptions need to be made about a flow in order to extend results about Ricci solitons. In this direction, we identify an identity, first exploited in the pioneering work of Richard Hamilton in the case of Ricci solitons, which we call Hamilton's identity. We show that a version of this identity for an arbitrary geometric flow allows one to recover results about rigidity, the growth of the potential function, volume growth and the Omori-Yau maximum principle that have been proven for gradient Ricci solitons.