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.

Appendix

Grönwall’s Inequality 6.7

Let u and f be continuous and non-negative functions defined on $I=[\alpha , \beta ]$, and let n(t) be a continuous, positive, nondecreasing function defined on I; then

$$\begin{aligned} u(t)\le n(t) + \int _{\alpha }^t f(s)u(s) \, \mathrm {d}s , \quad t \in I, \end{aligned}$$

(6.24)

implies that

$$\begin{aligned} u(t)\le n(t)\exp \left( \int _{\alpha }^t f(s) \, \mathrm {d}s\right) , \quad t \in I. \end{aligned}$$

(6.25)

Proof

See [9], theorem 1.3.1. $\square$