Removable singularity of positive mass theorem with continuous metrics

Abstract

In this paper, we consider asymptotically flat Riemannian manifolds endowed with a continuous metric and the metric is smooth away from a compact subset with certain conditions. If the metric is Lipschitz and the scalar curvature is nonnegative away from a closed subset with \((n-1)\)-dimensional Hausdorff measure zero, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that the manifold is isometric to the Euclidean space. The Hausdorff measure condition is optimal, which confirms a conjecture of Lee in 2013. The proof of the nonnegativity is based on a smoothing argument with some new estimates such as a \(L^1\)-scalar curvature approximation estimates. The proof of the rigidity relies on the RCD theory where we show that the manifold has nonnegative Ricci curvature in RCD sense. The idea involving RCD theory in the proof turns out to be much more suitable in the study of the rigidity for continuous metrics. Furthermore, our result generalizes the result of Lee and LeFloch (Commun Math Phys 339(1):99–120, 2015) from spin to non-spin.