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Transactions of the American Mathematical Society

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



Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume

Authors: F. Balacheff and S. Karam
Journal: Trans. Amer. Math. Soc. 372 (2019), 7071-7086
MSC (2010): Primary 53C23
Published electronically: March 26, 2019
MathSciNet review: 4024547
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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

S. Karam
Affiliation: Lebanese University, Beirut, Lebanon
MR Author ID: 1065486

Keywords: Guth conjecture, Schoen conjecture, smoothing inequality
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).
Article copyright: © Copyright 2019 American Mathematical Society