Positive scalar curvature and minimal hypersurface singularities
Positive scalar curvature and minimal hypersurface singularities
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DOI：
10.4310/sdg.2019.v24.n1.a10
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发表时间：
2017-04
期刊：
影响因子：
--
通讯作者：
R. Schoen;S. Yau
中科院分区：
文献类型：
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作者：
R. Schoen;S. Yau
在本文中，我们开发的方法来扩展的最小超曲面的方法，以正标量曲率问题的所有维度。这包括在没有自旋假设的情况下证明所有维度的正质量定理。它还包括声明的结构紧凑流形的正标量曲率扩展工作的\cite{sy 1}到所有维度。本文的技术工作是在小奇异集的存在下构造极小切片和相关的权函数，并证明在低维切片中奇异集不会变得太大。证明了任意切片上的奇异集是Hausdorff余维数至少为3的闭集。特别是对于涉及切片到维度$1$或$2$的参数，该方法是成功的。这些论点可以被看作是最小超曲面正则性理论在最小切片的设置上的扩展。
In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.