Mass under the Ricci flow

TL;DR

This paper investigates how the ADM mass of ALE spaces behaves under Ricci flow, proving invariance in three dimensions and implications for the flow's long-term behavior.

Contribution

It demonstrates that the ALE property is preserved under Ricci flow and shows the ADM mass remains invariant in three dimensions, with extensions to higher dimensions under certain conditions.

Findings

ALE property preserved under Ricci flow

ADM mass invariant in 3D Ricci flow

Non-zero mass prevents flow from converging to Euclidean space

Abstract

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let $(M,g)$ be an ALE manifold of dimension $n=3$. If $m(g)\neq 0$, then the Ricci flow starting at $g$ can not have Euclidean space as its (uniform) limit.