Geodesically complete spaces with an upper curvature bound

Geometric and Functional Analysis volume 29, pages295–342(2019)Cite this article

Abstract

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

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Acknowledgements

Most results of this paper have been obtained and presented in talks more than 10 years ago. The authors would like to express their gratitude to people, who have shown interest in our results and whose interest was responsible for the finalization of the paper. In particular, we would like to thank Werner Ballmann, Jérôme Bertrand, Pierre-Emmanuel Caprace, Karsten Grove, Vitali Kapovitch, Benoît Kloeckner, Urs Lang, Takao Yamaguchi. We are grateful to Anton Petrunin and Stephan Stadler for very helpful comments.

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Affiliations

  1. Mathematisches Institut, Universität Köln, Weyertal 86-90, 50937, Cologne, Germany

    Alexander Lytchak

  2. Institute of Mathematics, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan

    Koichi Nagano

Authors
  1. Alexander Lytchak
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  2. Koichi Nagano
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Correspondence to Alexander Lytchak.

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The first author was partially supported by the DFG grants SFB TRR 191 and SPP 2026. The second author was partially supported by JSPS KAKENHI Grant Numbers 26610012, 21740036, 18740023, and by the 2004 JSPS Postdoctoral Fellowships for Research Abroad

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Lytchak, A., Nagano, K. Geodesically complete spaces with an upper curvature bound. Geom. Funct. Anal. 29, 295–342 (2019). https://doi.org/10.1007/s00039-019-00483-7

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Keywords and phrases

  • Curvature bounds
  • Rectifiability
  • Metric measure spaces
  • Stratification
  • DC-functions

Mathematics Subject Classification

  • 53C20
  • 53C21
  • 53C23