An extension of Perelman's soul theorem for singular spaces

Details An extension of Perelman's soul theorem for singular spaces An extension of Perelman's soul theorem for singular spaces

2007-06-05

arxiv.org/abs/0706.0565v6

Mathematics

Differential Geometry

Update graphs, using the new trapezoid comparison theorem and new angular excess estimates to prove the main theorem

Scientific paper

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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