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.

中文翻译：

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具有分布曲率的非旋转流形的正质量定理

我们证明了具有分布曲率的流形的正质量定理，该定理已在Lee和LeFloch中进行了研究（Commun Math Phys 339（1）：99–120，2015），没有自旋条件。在我们的情况下，流形M具有渐近平坦的度量\（g \ in C ^ 0 \ bigcap W ^ {1，p} _ {-q} \），\（p> n \），\（q> \ frac {n-2} {2} \）。我们证明，只要\（q = n-2 \），广义ADM质量\（m_ {ADM}（M，g）\）都是非负的，并且g具有非负分布标量曲率，在Aleksandrov意义上为它的分布Ricci曲率属于一定的加权Lebesgue空间和一些额外条件。