Scalar curvature rigidity and Ricci Deturck flow on perturbations of Euclidean Space

TL;DR

This paper establishes a rigidity result for non-negative scalar curvature perturbations of Euclidean space using Ricci DeTurck flow, extending known bounds and analyzing flow regularity at initial data.

Contribution

It introduces a new rigidity theorem for scalar curvature perturbations and advances understanding of Ricci DeTurck flow behavior on Euclidean perturbations.

Findings

Proves a weak version of the positive mass theorem rigidity.

Extends $L^p$ bounds and decay rates for Ricci DeTurck flow.

Establishes flow regularity at initial data.

Abstract

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known $L^p$ bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.