具有分布曲率流形的正质量定理
The positive mass theorem for manifolds with distributional curvature
摘要：
本文给出并证明了度量只有Sobolev正则性$C^0 \cap W^{1,n}$的n维自旋流形的正质量定理。在这个层次上的规则性，曲率的度量定义的分布意义上，我们在这里提出了一个（广义）的概念ADM质量这样一个度量。我们的主要定理建立了，如果流形是渐近平坦的，并有非负的标量曲率分布，那么它的（广义）ADM质量是明确的和非负的，只有当流形是等距欧几里德空间为零。李和帕克以及巴特尼克在以前应用威滕旋量方法时，需要更强的正则性$W^{2,2}$。我们的证明是一个推广威滕的论点，其中我们必须处理狄拉克算子及其相关的Lichnerowicz-Weitzenbock身份在分布意义上，并应付一定的平均值的一阶导数的度量超过annuli，接近无穷大。最后，我们观察到，我们的论点不是特定的标量曲率，也允许我们建立一个普遍的正质量定理。
英文摘要：
We formulate and prove a positive mass theorem for n-dimensional spin manifolds whose metrics have only the Sobolev regularity $C^0 \cap W^{1,n}$. At this level of regularity, the curvature of the metric is defined in the distributional sense only, and we propose here a (generalized) notion of ADM mass for such a metric. Our main theorem establishes that if the manifold is asymptotically flat and has non-negative scalar curvature distribution, then its (generalized) ADM mass is well-defined and non-negative, and vanishes only if the manifold is isometric to Euclidian space. Prior applications of Witten's spinor method by Lee and Parker and by Bartnik required the much stronger regularity $W^{2,2}$. Our proof is a generalization of Witten's arguments, in which we must treat the Dirac operator and its associated Lichnerowicz-Weitzenbock identity in the distributional sense and cope with certain averages of first-order derivatives of the metric over annuli that approach infinity. Finally, we observe that our arguments are not specific to scalar curvature and also allow us to establish a universal positive mass theorem.
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