Yamabe flow and ADM Mass on asymptotically flat manifolds

TL;DR

This paper studies how the ADM mass and Einstein-Hilbert functional behave under the Yamabe flow on asymptotically flat manifolds, showing monotonicity and invariance properties in certain dimensions.

Contribution

It demonstrates that ADM mass and Einstein-Hilbert functional are well-defined and monotone under Yamabe flow, and shows invariance of ADM mass in 3 and 4 dimensions.

Findings

ADM mass is well-defined and monotone under Yamabe flow.

In 3 and 4 dimensions, ADM mass remains invariant during the flow.

Yamabe flow acts as the gradient flow of Einstein-Hilbert functional in these cases.

Abstract

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifolds