The Positive Mass Theorem for Non-spin Manifolds with Distributional Curvature

We prove the positive mass theorem for manifolds with distributional curvature which have been studied in Lee and LeFloch (Commun Math Phys 339(1):99–120, 2015) 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 nonnegative as long as $$q=n-2$$, and g has nonnegative distributional scalar curvature, bounded curvature in the Aleksandrov sense with its distributional Ricci curvature belonging to certain weighted Lebesgue space and some extra conditions.