CD meets CAT

Vitali Kapovitch; Christian Ketterer

doi:10.1515/crelle-2019-0021

Article

CD meets CAT

Vitali Kapovitch and Christian Ketterer

Published/Copyright: August 20, 2019

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 766

Abstract

We show that if a noncollapsed CD ⁢ ( K , n ) space X with n ≥ 2 has curvature bounded above by κ in the sense of Alexandrov, then K ≤ ( n - 1 ) ⁢ κ and X is an Alexandrov space of curvature bounded below by K - κ ⁢ ( n - 2 ) . We also show that if a CD ⁢ ( K , n ) space Y with finite n has curvature bounded above, then it is infinitesimally Hilbertian.

Funding statement: The first author was supported in part by a Discovery grant from NSERC. This work was done while the second author was participating in Fields Thematic Program on “Geometric Analysis” from July till December 2017. Both authors want to thank the Fields Institute for providing an excellent and stimulating research environment.

Acknowledgements

The authors are grateful to Robert Haslhofer for helpful conversations and comments. The authors also want to thank Nicola Gigli for comments and remarks that helped to improve an earlier version of this article. Lastly, the authors are also grateful to the referee for several useful comments and suggestions.

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Received: 2018-04-13

Revised: 2019-03-20

Published Online: 2019-08-20

Published in Print: 2020-09-01

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https://doi.org/10.1515/crelle-2019-0021