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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume
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by F. Balacheff and S. Karam PDF
Trans. Amer. Math. Soc. 372 (2019), 7071-7086 Request permission

Abstract:

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic one. This result is deduced from a nonsharp macroscopic version of a conjecture by R. Schoen about scalar curvature, whose proof is a variation of an argument due to M. Gromov and is based on a smoothing technique. We take the opportunity of this work to present a full account of this technique, which involves simplicial volume and deserves to be better known.
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Additional Information
  • F. Balacheff
  • Affiliation: Universitat Autònoma de Barcelona, Barcelona, Spain
  • MR Author ID: 759115
  • ORCID: 0000-0001-9770-2954
  • Email: fbalacheff@mat.uab.cat
  • S. Karam
  • Affiliation: Lebanese University, Beirut, Lebanon
  • MR Author ID: 1065486
  • Email: karam.steve.work@gmail.com
  • Received by editor(s): July 8, 2018
  • Received by editor(s) in revised form: November 21, 2018
  • Published electronically: March 26, 2019
  • Additional Notes: The first author acknowledges support from grants ANR Finsler (ANR-12-BS01-0009-02) and Ramón y Cajal (RYC-2016-19334).
    The second author acknowledges support from grant ANR CEMPI (ANR-11-LABX-0007-01).
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7071-7086
  • MSC (2010): Primary 53C23
  • DOI: https://doi.org/10.1090/tran/7765
  • MathSciNet review: 4024547