Scalar curvature lower bounds on asymptotically flat manifolds

TL;DR

This paper investigates scalar curvature lower bounds on asymptotically flat manifolds with $W^{1,p}$ metrics, establishing their equivalence at infinity and analyzing their behavior under Ricci-DeTurck flow.

Contribution

It demonstrates the relationship between different notions of scalar curvature lower bounds on asymptotically flat manifolds and their evolution under Ricci-DeTurck flow.

Findings

Scalar curvature lower bounds depend on the flow and time.

Distributional and $eta$-weak scalar curvature bounds coincide at infinity.

Lower bounds are preserved and related under Ricci-DeTurck flow.

Abstract

In this paper, we consider the scalar curvature in the distributional sense of \cite{MR3366052} and the scalar curvature lower bound in the $\beta-$weak $(\beta\in(0, \frac{1}{2}))$ sense of \cite{MR4685089} on an asymptotically flat $n-$manifold with a $W^{1,p}(p>n)$ metric. We first show that the scalar curvature lower bound under the Ricci-DeTurck flow depends on the scalar curvature lower bound in the $\beta-$weak sense and the time. Then we prove that the lower bound of the distributional scalar curvature of a $W^{1, p}$ metric coincides with the lower bound of the scalar curvature in the $\beta-$weak sense at infinity.