The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

TL;DR

This paper investigates the behavior of the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant, showing convergence after rescaling to a solution of the Yamabe problem on a compactification.

Contribution

It demonstrates that, unlike the divergence in the non-positive case, the Yamabe flow converges after rescaling to a unique solution on the compactified manifold.

Findings

Yamabe flow diverges without rescaling when Y ≤ 0

Rescaled flow converges to the Yamabe problem solution

Convergence occurs on a compactification of the original manifold

Abstract

We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weighted sense when $Y>0$, the flow must diverge when $Y\leq 0$. We show here in the $Y\leq 0$ case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.