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

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 in the metric-space sense is obtained via the volume comparison theory of $\mathsf{RCD}$-spaces after establishing that zero ADM mass implies Ricci flatness, which fundamentally relies on these improved estimates. 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.

36 pages; revised and expanded the rigidity section, with a more general result; improved presentation and minor changes