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arXiv · 0706.0565

An extension of Perelman's soul theorem for singular spaces

Jianguo Cao·Bo Dai·Jiaqiang Mei
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Abstract

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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Jianguo Cao, Bo Dai, Jiaqiang Mei. 2010-06-09. An extension of Perelman's soul theorem for singular spaces. https://arxiv.org/abs/0706.0565

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