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

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}$ 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 $C^0\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$ metrics with $p>n$ via the comparison theory of $\mathsf{R}\mathsf{C}\mathsf{D}$-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.