A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature

TL;DR

This paper proves a positive mass theorem for non-spin manifolds with low-regularity metrics and distributional curvature, extending classical results to less smooth settings using approximation and Sobolev techniques.

Contribution

It establishes a positive mass theorem for non-spin manifolds with metrics of low regularity and distributional scalar curvature, removing the spin condition under certain smoothness assumptions.

Findings

Asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass.

Rigidity holds for certain low-regularity metrics with $p>n$ via $ ext{RCD}$-space theory.

The theorem generalizes Lee-LeFloch's result by removing the spin condition for metrics smooth outside a compact set.

Abstract

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for $C^0 \cap W_{\mathrm{loc}}^{1,p}$ metrics with $p>n$ via the comparison theory of $\sf{RCD}$-spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set.