An extension of Perelman's soul theorem for singular spaces
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.
Keywords
Cite
@article{arxiv.0706.0565,
title = {An extension of Perelman's soul theorem for singular spaces},
author = {Jianguo Cao and Bo Dai and Jiaqiang Mei},
journal= {arXiv preprint arXiv:0706.0565},
year = {2010}
}
Comments
Update graphs, using the new trapezoid comparison theorem and new angular excess estimates to prove the main theorem