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Transactions of the American Mathematical Society

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Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume


Authors: F. Balacheff and S. Karam
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 53C23
DOI: https://doi.org/10.1090/tran/7765
Published electronically: March 26, 2019
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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
Email: fbalacheff@mat.uab.cat

S. Karam
Affiliation: Lebanese University, Beirut, Lebanon
Email: karam.steve.work@gmail.com

DOI: https://doi.org/10.1090/tran/7765
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