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

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.