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 $C^0 \cap W_{\mathrm{loc}}^{1,n}$ 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.

💡 Research Summary

The paper establishes a low‑regularity version of the Riemannian positive mass theorem for non‑spin manifolds whose metrics belong only to the class (C^{0}\cap W^{1,n}_{\text{loc}}) and are smooth outside a compact set. The main result (Theorem 1.2) asserts two statements. First, if ((M^{n},g)) is complete, asymptotically flat, and its distributional scalar curvature (R