Riemannian hyperbolization

Pedro Ontaneda

doi:10.1007/s10240-020-00113-1

Article

Riemannian hyperbolization

Pedro Ontaneda

Publications Mathématiques de l'IHÉS, Volume 131 (2020), pp. 1-72

Abstract

The strict hyperbolization process of Charney and Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by Gromov and later studied by Davis and Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is very far from being Riemannian because it has a large and complicated set of singularities. We show that these singularities can be removed (provided the hyperolization piece is large). Hence the strict hyperbolization process can be done in the Riemannian setting.

Received: 2016-09-30
Accepted: 2019-12-30
Online First: 2020-02-28
Published online: 2020-06-07

MR Zbl

DOI: 10.1007/s10240-020-00113-1

@article{PMIHES_2020__131__1_0,
     author = {Pedro Ontaneda},
     title = {Riemannian hyperbolization},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--72},
     year = {2020},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {131},
     doi = {10.1007/s10240-020-00113-1},
     mrnumber = {4106793},
     zbl = {1442.53026},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-020-00113-1/}
}

TY  - JOUR
AU  - Pedro Ontaneda
TI  - Riemannian hyperbolization
JO  - Publications Mathématiques de l'IHÉS
PY  - 2020
SP  - 1
EP  - 72
VL  - 131
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-020-00113-1/
DO  - 10.1007/s10240-020-00113-1
LA  - en
ID  - PMIHES_2020__131__1_0
ER  -

%0 Journal Article
%A Pedro Ontaneda
%T Riemannian hyperbolization
%J Publications Mathématiques de l'IHÉS
%D 2020
%P 1-72
%V 131
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-020-00113-1/
%R 10.1007/s10240-020-00113-1
%G en
%F PMIHES_2020__131__1_0

Pedro Ontaneda. Riemannian hyperbolization. Publications Mathématiques de l'IHÉS, Volume 131 (2020), pp. 1-72. doi: 10.1007/s10240-020-00113-1

References
Cited by

[1.] S. Ardanza-Trevillano, Exotic smooth structures on negatively curved manifolds that are not of the homotopy type of a locally symmetric space, PhD. Thesis, SUNY Binghamton, 2000. | MR

[2.] I. Belegradek; V. Kapovitch Classification of negatively pinched manifolds with amenable fundamental groups, Acta Math., Volume 196 (2006), pp. 229-260 see also corrected version (28 August 2010), arXiv:math1040 2268rt | MR | DOI | Zbl

[3.] R. L. Bishop; B. O’Neill Manifolds of negative curvature, Trans. Am. Math. Soc., Volume 145 (1969), pp. 1-49 | MR | DOI | Zbl

[4.] M. Bridson; A. Haeflinger Metric spaces of non-positive curvature, Springer, Berlin, 1999 | DOI | MR | Zbl

[5.] R. M. Charney; M. W. Davis Strict hyperbolization, Topology, Volume 34 (1995), pp. 329-350 | MR | Zbl | DOI

[6.] K. Corlette Archimedean superrigidity and hyperbolic geometry, Ann. Math., Volume 135 (1992), pp. 165-182 | MR | DOI | Zbl

[7.] J. F. Davis; F. Fang An almost flat manifolds with a cyclic or quaternionic holonomy group bounds, J. Differ. Geom., Volume 103 (2016), pp. 289-296 | MR | Zbl | DOI

[8.] M. W. Davis; T. Januszkiewicz Hyperbolization of polyhedra, J. Differ. Geom., Volume 34 (1991), pp. 347-388 | MR | DOI | Zbl

[9.] M. Deraux A negatively curved Kähler threefold not covered by the ball, Invent. Math., Volume 160 (2005), pp. 501-525 | MR | Zbl | DOI

[10.] J. Eells; J. H. Sampson Harmonic mappings of Riemannian manifolds, Am. J. Math., Volume 86 (1964), pp. 109-160 | MR | DOI | Zbl

[11.] F. T. Farrell; W. C. Hsiang The Whitehead group of poly-(finite or cyclic) groups, J. Lond. Math. Soc., Volume 24 (1982), pp. 308-324 | MR | Zbl

[12.] F. T. Farrell; L. E. Jones Negatively curved manifolds with exotic smooth structures, J. Am. Math. Soc., Volume 2 (1989), pp. 899-908 | MR | DOI | Zbl

[13.] F. T. Farrell; S. Zdravskoska Do almost flat manifolds bound?, Mich. Math. J., Volume 30 (1983), pp. 199-208 | MR | Zbl | DOI

[14.] M. Gromov Manifolds of negative curvature, J. Differ. Geom., Volume 13 (1978), pp. 231-241 | MR | DOI

[15.] M. Gromov Almost flat manifold, J. Differ. Geom., Volume 13 (1978), pp. 223-230 | MR | DOI | Zbl

[16.] M. Gromov Hyperbolic groups (S. M. Gersten, ed.), Essays in group theory, 8, Springer, New York, 1987, pp. 75-284 | MR | Zbl | DOI

[17.] M. Gromov Foliated plateau problem, Geom. Funct. Anal., Volume 1 (1991), pp. 14-79 | MR | DOI | Zbl

[18.] M. Gromov; W. Thurston Pinching constants for hyperbolic manifolds, Invent. Math., Volume 89 (1987), pp. 1-12 | MR | Zbl | DOI

[19.] G. C. Hamrick; D. C. Royster Flat Riemannian manifolds are boundaries, Invent. Math., Volume 66 (1982), pp. 405-413 | MR | DOI | Zbl

[20.] L. Hernández Kähler manifolds and 1/4-pinching, Duke Math. J., Volume 62 (1991), pp. 601-611 | MR | DOI | Zbl

[21.] D. D. Long; A. W. Reid On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topol., Volume 4 (2000), pp. 171-178 | MR | DOI | Zbl

[22.] N. Mok; Y.-T. Siu; S.-K. Yeung Geometric superrigidity, Invent. Math., Volume 113 (1993), pp. 57-83 | MR | DOI | Zbl

[23.] G. D. Mostow; Y. T. Siu A compact Kähler manifold of negative curvature not covered by the ball, Ann. Math., Volume 112 (1980), pp. 312-360 | DOI | MR | Zbl

[24.] J. R. Munkres Elementary differential topology, 54, Princeton University Press, Princeton, 1963 | DOI | MR

[25.] S. P. Novikov Rational Pontryagin classes. Homeomorphism and homotopy type of closed manifolds. I, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 29 (1965), pp. 1373-1388 | MR | Zbl

[26.] Olshanski Almost every group is hyperbolic, Int. J. Algebra Comput., Volume 2 (1992), pp. 1-17 | MR | DOI | Zbl

[27.] P. Ontaneda On the Farrell and Jones warping deformation, J. Lond. Math. Soc., Volume 92 (2015), pp. 566-582 | MR | DOI | Zbl

[28.] P. Ontaneda Normal smoothings for smooth cube manifolds, Asian J. Math., Volume 20 (2016), pp. 709-724 | MR | Zbl | DOI

[29.] P. Ontaneda Deforming an $ε$ -close to hyperbolic metric to a warp product, Mich. Math. J., Volume 65 (2016), pp. 293-701 | MR | Zbl | DOI

[30.] P. Ontaneda Cut limits on hyperbolic extensions, Mich. Math. J., Volume 65 (2016), pp. 703-714 | MR | DOI | Zbl

[31.] P. Ontaneda Normal smoothings for Charney-Davis strict hyperbolizations, J. Topol. Anal., Volume 9 (2017), pp. 127-165 | MR | DOI | Zbl

[32.] P. Ontaneda Hyperbolic extensions and metrics $ε$ -close to hyperbolic, Indiana Univ. Math. J., Volume 66 (2017), pp. 609-630 | MR | Zbl | DOI

[33.] P. Ontaneda Deforming an $ε$ -close to hyperbolic metric to a hyperbolic metric, Proc. R. Soc. Edinb. A, Volume 148 (2018), pp. 629-641 | MR | Zbl | DOI

[34.] E. A. Ruh Almost flat manifolds, J. Differ. Geom., Volume 17 (1982), pp. 1-14 | MR | DOI | Zbl

[35.] J. Sampson Applications of harmonic maps to Kähler geometry, Contemp. Math., Volume 49 (1986), pp. 125-133 | DOI | MR | Zbl

[36.] Z. M. Shen On complete Riemannian manifolds with collapsed ends, Pac. J. Math., Volume 163 (1994), pp. 175-182 | MR | DOI | Zbl

[37.] S. Upadhyay A bounding question for almost flat manifolds, Trans. Am. Math. Soc., Volume 353 (2001), pp. 963-972 | MR | DOI | Zbl

[38.] S.-T. Yau Open problems in differential geometry, Proc. Symp. Pure Math., Volume 54 (1993), pp. 1-28 | Zbl | MR

[39.] F. Zheng Hirzebruch-Kato surfaces, Deligne-Mostow’s construction, and new examples of negatively curved compact Kähler surfaces, Invent. Math., Volume 103 (1991), pp. 527-535 | MR | DOI

Cited by Sources: