The positive mass theorem for non-spin manifolds with distributional curvature

TL;DR

This paper extends the positive mass theorem to non-spin manifolds with distributional curvature, establishing non-negativity of ADM mass under specific regularity and curvature conditions.

Contribution

It proves the positive mass theorem for non-spin manifolds with distributional curvature, relaxing previous spin condition requirements.

Findings

ADM mass is non-negative for the class of manifolds considered.

The theorem applies to manifolds with asymptotically flat metrics in certain Sobolev spaces.

Distributional scalar curvature and Ricci curvature conditions are sufficient for the positive mass theorem.

Abstract

We prove the positive mass theorem for manifolds with distributional curvature which have been studied in \cite{Lee2015} without spin condition. In our case, the manifold $M$ has asymptotically flat metric $g\in C^0\bigcap W^{1,p}_{-q}$, $p>n$, $q>\frac{n-2}{2}$. We show that the generalized ADM mass $m_{ADM}(M,g)$ is non-negative as long as $q=n-2$, and $g$ has non-negative distributional scalar curvature, bounded curvature in the Alexandrov sense with its distributional Ricci curvature belonging to certain weighted Lebesgue space and some extra conditions.