Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume
HTML articles powered by AMS MathViewer
- 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.References
- G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), no. 5, 731–799 (French). MR 1354289, DOI 10.1007/BF01897050
- Guillaume Bulteau, Cycles géométriques réguliers, Bull. Soc. Math. France 143 (2015), no. 4, 727–761 (French, with English and French summaries). MR 3450500, DOI 10.24033/bsmf.2703
- Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287, DOI 10.24033/asens.1390
- Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
- Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
- Mikhael Gromov, Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 291–362 (English, with English and French summaries). MR 1427763
- Mikhael Gromov and H. Blaine Lawson Jr., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2) 111 (1980), no. 2, 209–230. MR 569070, DOI 10.2307/1971198
- M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12. MR 892185, DOI 10.1007/BF01404671
- Larry Guth, Metaphors in systolic geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 745–768. MR 2827817
- Larry Guth, Volumes of balls in large Riemannian manifolds, Ann. of Math. (2) 173 (2011), no. 1, 51–76. MR 2753599, DOI 10.4007/annals.2011.173.1.2
- Steve Karam, Growth of balls in the universal cover of surfaces and graphs, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5355–5373. MR 3347175, DOI 10.1090/S0002-9947-2015-06189-3
- G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383, DOI 10.1007/BF02684590
- Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
- R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700, DOI 10.1007/BF01647970
- W. P. Thurston, Geometry and topology of $3$-manifolds, Princeton University Press, Princeton, NJ, 1978.
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