Riemannian Positive Mass Theorem in All Dimensions in the Presence of Low-Codimension Singularities

Abstract: We prove the Riemannian positive mass theorem in all dimensions for asymptotically flat $L<sup>\infty$-metrics with subcritical singular sets. More precisely, we consider complete asymptotically flat manifolds whose metrics are smooth away from a compact singular set of Minkowski dimension less than $n-3+\frac{2}{n}$, and whose scalar curvature is nonnegative on the regular set. We show that the ADM mass of each asymptotically flat end is nonnegative, and that the mass vanishes in some end only in the Euclidean case. For the rigidity statement, we require additionally that the Minkowski dimension of the singular set is not larger than $n-3+\frac{1}{n-1}$. This gives an asymptotically flat analogue of Schoen's codimension-three conjecture for positive scalar curvature. The proof combines a density theorem for singular asymptotically flat metrics, capacity estimates across the singular set, conformal blow-up inspired by Bi-Hao-He-Shi-Zhu [3], and a $μ$-bubble dimension-descent argument adapted from Brendle-Wang [6].