A renormalized Perelman-functional and a lower bound for the ADM-mass

TL;DR

This paper introduces a renormalized Perelman functional for perturbations of steady Ricci solitons and defines a geometric invariant that provides a lower bound for the ADM-mass, linking Ricci flow stability and general relativity.

Contribution

It proposes a new renormalized functional for Ricci solitons and a geometric invariant that bounds the ADM-mass, connecting Ricci flow analysis with gravitational mass concepts.

Findings

A stability inequality for Ricci solitons is established.

A new invariant lambda_AF provides a lower bound for the ADM-mass.

Discovery of a mass decreasing flow in three dimensions.

Abstract

In the first part of this short article, we define a renormalized F-functional for perturbations of non-compact steady Ricci solitons. This functional motivates a stability inequality which plays an important role in questions concerning the regularity of Ricci-flat spaces and the non-uniqueness of the Ricci flow with conical initial data. In the second part, we define a geometric invariant lambda_AF for asymptotically flat manifolds with nonnegative scalar curvature. This invariant gives a quantitative lower bound for the ADM-mass from general relativity, motivates a Ricci flow proof of the rigidity statement in the positive mass theorem, and eventually leads to the discovery of a mass decreasing flow in dimension three.