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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Rigidity of nonpositively curved manifolds with convex boundary
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by Mohammad Ghomi and Joel Spruck
Proc. Amer. Math. Soc. 151 (2023), 4935-4940
DOI: https://doi.org/10.1090/proc/16475
Published electronically: July 21, 2023

Abstract:

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv a$ on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for $K\geq a\geq 0$.
References
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Bibliographic Information
  • Mohammad Ghomi
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 687341
  • ORCID: 0000-0003-3767-9347
  • Email: ghomi@math.gatech.edu
  • Joel Spruck
  • Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
  • MR Author ID: 165780
  • ORCID: 0009-0003-0609-8029
  • Email: js@math.jhu.edu
  • Received by editor(s): October 30, 2022
  • Received by editor(s) in revised form: October 30, 2022, and March 3, 2023
  • Published electronically: July 21, 2023
  • Additional Notes: The research of the first author was supported by NSF grant DMS-2202337 and a Simons Fellowship. The research of the second author was supported by a Simons Collaboration Grant
  • Communicated by: Jiaping Wang
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 4935-4940
  • MSC (2020): Primary 53C20, 58J05; Secondary 53C42, 52A15
  • DOI: https://doi.org/10.1090/proc/16475
  • MathSciNet review: 4634894