The Yamabe flow on asymptotically flat manifolds

TL;DR

This paper investigates the behavior of the Yamabe flow on asymptotically flat manifolds, demonstrating convergence to scalar-flat metrics when the Yamabe invariant is positive and analyzing the ADM mass evolution.

Contribution

It establishes conditions for convergence of the Yamabe flow on asymptotically flat manifolds and links the ADM mass change to scalar curvature limits.

Findings

Flow converges to scalar-flat metric if Y(M,[g_0])>0

Flow does not converge if Y(M,[g_0])≤0

ADM mass decreases by total scalar curvature limit

Abstract

We study the Yamabe flow starting from an asymptotically flat manifold $(M^n,g_0)$. We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if $Y(M,[g_0])>0$, and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.