Perelman灵魂定理在奇异空间中的推广
An extension of Perelman's soul theorem for singular spaces
摘要：
本文研究具有非负曲率的开完全度量空间。其中，我们建立了Perelman灵魂定理在可能奇异空间中的推广：“设X是一个完备的、非紧的、有限维的Alexandrov空间，具有非负曲率。设X在非空开子集上无边界且有正曲率。则X一定是一个可缩空间“。这个结果的证明使用了距离函数和Busemann函数在具有非负曲率的奇异空间上的性质的详细分析。我们将引入一类角过剩函数来度量奇异空间中凸超曲面的凸性和外曲率。我们还得到了一个新的比较，在非负弯曲的空间，这导致了我们的新的灵魂定理的证明所需的凸性估计。
英文摘要：
In this paper, we study open complete metric spaces with non-negative curvature. Among other things, we establish an extension of Perelman's soul theorem for possibly singular spaces: "Let X be a complete, non-compact, finite dimensional Alexandrov space with non-negative curvature. Suppose that X has no boundary and has positive curvature on a non-empty open subset. Then X must be a contractible space". The proof of this result uses the detailed analysis of concavity of distance functions and Busemann functions on singular spaces with non-negative curvature. We will introduce a family of angular excess functions to measure convexity and extrinsic curvature of convex hypersurfaces in singular spaces. We also derive a new comparison for trapezoids in non-negatively curved spaces, which led to desired convexity estimates for the proof of our new soul theorem.
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