A cyclic extension  of the earthquake flow II

Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc

doi:10.24033/asens.2259

A cyclic extension of the earthquake flow II
[Une extension cyclique du flot des tremblements de terre II]

Bonsante, Francesco ; Mondello, Gabriele ; Schlenker, Jean-Marc

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 811-859.

Résumé
Abstract

Le flot des glissements de terrain, introduit dans [5], est un analogue régulier du flot des tremblements de terre sur l'espace de Teichmüller, qui partage certaines de ses principales propriétés. Nous montrons ici que d'autres propriétés des tremblements de terre s'appliquent aux glissements de terrain. Le flot des glissements de terrain est le flot hamiltonien d'une fonction convexe. L'application de greffage régulière $sgr$ , à valeur dans l'espace de Teichmüller, qui est aux glissements de terrain ce que le greffage est aux tremblements de terre, est propre et surjective par rapport à chacune de ses variables. L'application de greffage régulière $SGr$ , à valeur dans l'espace des structures projectives complexes, est symplectique (à un facteur multiplicatif près). La composition de deux glissements de terrain a un point fixe dans l'espace de Teichmüller. En conséquence, nous obtenons des résultats nouveaux sur les surfaces à courbure de Gauss constantes dans des variétés de dimension 3 hyperboliques ou AdS. Nous montrons aussi que le flot des glissements de terrain a une extension satisfaisante au bord de l'espace de Teichmüller.

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichmüller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichmüller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.

Publié le : 2015-09-24

DOI : 10.24033/asens.2259

Classification : 30F60, 53C42, 49Q10, 53C43.
Keywords: Espace de Teichmüller, surfaces hyperboliques, applications minimales lagrangiennes, métrique de Weil-Petersson, tenseurs de Codazzi.
Mot clés : Teichmüller space, hyperbolic surfaces, minimal Lagrangian maps, Weil-Petersson metric, Codazzi tensors.

@article{ASENS_2015__48_4_811_0,
     author = {Bonsante, Francesco and Mondello, Gabriele and Schlenker, Jean-Marc},
     title = {A cyclic extension  of the earthquake {flow~II}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {811--859},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {4},
     year = {2015},
     doi = {10.24033/asens.2259},
     mrnumber = {3377066},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2259/}
}

TY  - JOUR
AU  - Bonsante, Francesco
AU  - Mondello, Gabriele
AU  - Schlenker, Jean-Marc
TI  - A cyclic extension  of the earthquake flow II
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2015
SP  - 811
EP  - 859
VL  - 48
IS  - 4
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2259/
DO  - 10.24033/asens.2259
LA  - en
ID  - ASENS_2015__48_4_811_0
ER  -

%0 Journal Article
%A Bonsante, Francesco
%A Mondello, Gabriele
%A Schlenker, Jean-Marc
%T A cyclic extension  of the earthquake flow II
%J Annales scientifiques de l'École Normale Supérieure
%D 2015
%P 811-859
%V 48
%N 4
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2259/
%R 10.24033/asens.2259
%G en
%F ASENS_2015__48_4_811_0

Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc. A cyclic extension  of the earthquake flow II. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 811-859. doi : 10.24033/asens.2259. http://www.numdam.org/articles/10.24033/asens.2259/

Bibliographie
Cité par

Andersson, L.; Barbot, T.; Benedetti, R.; Bonsante, F.; Goldman, W. M.; Labourie, F.; Scannell, K. P.; Schlenker, J.-M. Notes on: “Lorentz spacetimes of constant curvature” [Geom. Dedicata 126 (2007), 3–45] by G. Mess, Geom. Dedicata, Volume 126 (2007), pp. 47-70 (ISSN: 0046-5755) | DOI | MR

Ahlfors, L. V. Some remarks on Teichmüller's space of Riemann surfaces, Ann. of Math., Volume 74 (1961), pp. 171-191 (ISSN: 0003-486X) | DOI | MR | Zbl

Benedetti, R.; Bonsante, F. Canonical Wick rotations in 3-dimensional gravity, Mem. Amer. Math. Soc., Volume 198 (2009) (ISBN: 978-0-8218-4281-2, ISSN: 0065-9266) | DOI | MR | Zbl

Barbot, T.; Bonsante, F.; Danciger, J.; Goldman, W. M.; Guéritaud, F.; Kassel, F.; Krasnov, K.; Schlenker, J.-M.; Zeghib, A. Some open questions on anti-de Sitter geometry (preprint arXiv:1205.6103 )

Barbot, T.; Bonsante, F.; Schlenker, J.-M. Collisions of particles in locally AdS spacetimes I. Local description and global examples, Comm. Math. Phys., Volume 308 (2011), pp. 147-200 (ISSN: 0010-3616) | DOI | MR | Zbl

Barbot, T.; Béguin, F.; Zeghib, A. Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble), Volume 61 (2011), pp. 511-591 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Bers, L.; Ehrenpreis, L. Holomorphic convexity of Teichmüller spaces, Bull. Amer. Math. Soc., Volume 70 (1964), pp. 761-764 (ISSN: 0002-9904) | DOI | MR | Zbl

Bonsante, F.; Mondello, G.; Schlenker, J.-M. A cyclic extension of the earthquake flow I, Geom. Topol., Volume 17 (2013), pp. 157-234 (ISSN: 1465-3060) | DOI | MR | Zbl

Bonsante, F.; Schlenker, J.-M. Fixed points of compositions of earthquakes, Duke Math. J., Volume 161 (2012), pp. 1011-1054 (ISSN: 0012-7094) | DOI | MR | Zbl

Casson, A.; Jungreis, D. Convergence groups and Seifert fibered 3-manifolds, Invent. Math., Volume 118 (1994), pp. 441-456 (ISSN: 0020-9910) | DOI | MR | Zbl

Diallo, B. Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compact ads 3-manifolds (preprint arXiv:1303.7406 )

Dumas, D.; Wolf, M. Projective structures, grafting and measured laminations, Geom. Topol., Volume 12 (2008), pp. 351-386 (ISSN: 1465-3060) | DOI | MR | Zbl

Eells, J.; Lemaire, L. Deformations of metrics and associated harmonic maps, Proc. Indian Acad. Sci. Math. Sci., Volume 90 (1981), pp. 33-45 (ISSN: 0370-0089) | DOI | MR | Zbl

Fathi, A.; Laudenbach, F.; Poenaru, V., Astérisque, 66–67, Soc. Math. France, 1991 | Numdam | MR | Zbl

Fischer, A. E.; Tromba, A. J. A new proof that Teichmüller space is a cell, Trans. Amer. Math. Soc., Volume 303 (1987), pp. 257-262 (ISSN: 0002-9947) | DOI | MR | Zbl

Gabai, D. Convergence groups are Fuchsian groups, Ann. of Math., Volume 136 (1992), pp. 447-510 (ISSN: 0003-486X) | DOI | MR | Zbl

Gardiner, F. P. Measured foliations and the minimal norm property for quadratic differentials, Acta Math., Volume 152 (1984), pp. 57-76 (ISSN: 0001-5962) | DOI | MR | Zbl

Graham, C. R.; Witten, E. Conformal anomaly of submanifold observables in AdS/CFT correspondence (preprint arXiv:hep-th/9901021 ) | MR | Zbl

Hubbard, J.; Masur, H. Quadratic differentials and foliations, Acta Math., Volume 142 (1979), pp. 221-274 (ISSN: 0001-5962) | DOI | MR | Zbl

Hodgson, C. D.; Rivin, I. A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., Volume 111 (1993), pp. 77-111 (ISSN: 0020-9910) | DOI | MR | Zbl

Kawai, S. The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann., Volume 305 (1996), pp. 161-182 (ISSN: 0025-5831) | DOI | MR | Zbl

Kerckhoff, S. P. The Nielsen realization problem, Ann. of Math., Volume 117 (1983), pp. 235-265 (ISSN: 0003-486X) | DOI | MR | Zbl

Kerckhoff, S. P. Lines of minima in Teichmüller space, Duke Math. J., Volume 65 (1992), pp. 187-213 (ISSN: 0012-7094) | DOI | MR | Zbl

Krasnov, K.; Schlenker, J.-M. Minimal surfaces and particles in 3-manifolds, Geom. Dedicata, Volume 126 (2007), pp. 187-254 (ISSN: 0046-5755) | DOI | MR | Zbl

Krasnov, K.; Schlenker, J.-M. On the renormalized volume of hyperbolic 3-manifolds, Comm. Math. Phys., Volume 279 (2008), pp. 637-668 (ISSN: 0010-3616) | DOI | MR | Zbl

Krasnov, K.; Schlenker, J.-M. A symplectic map between hyperbolic and complex Teichmüller theory, Duke Math. J., Volume 150 (2009), pp. 331-356 (ISSN: 0012-7094) | DOI | MR | Zbl

Krasnov, K.; Schlenker, J.-M., Handbook of Teichmüller theory. Volume III (IRMA Lect. Math. Theor. Phys.), Volume 17, Eur. Math. Soc., Zürich, 2012, pp. 779-819 | DOI | MR | Zbl

Kamishima, Y.; Tan, S. P., Aspects of low-dimensional manifolds (Adv. Stud. Pure Math.), Volume 20, Kinokuniya, Tokyo, 1992, pp. 263-299 | DOI | MR | Zbl

Labourie, F. Surfaces convexes dans l'espace hyperbolique et $𝐂 P^{1}$ -structures, J. London Math. Soc., Volume 45 (1992), pp. 549-565 (ISSN: 0024-6107) | DOI | MR | Zbl

McMullen, C. T. Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc., Volume 11 (1998), pp. 283-320 (ISSN: 0894-0347) | DOI | MR | Zbl

Mess, G. Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 (ISSN: 0046-5755) | DOI | MR | Zbl

Papadopoulos, A.; Penner, R. C. The Weil-Petersson symplectic structure at Thurston's boundary, Trans. Amer. Math. Soc., Volume 335 (1993), pp. 891-904 (ISSN: 0002-9947) | DOI | MR | Zbl

Papadopoulos, A.; Théret, G. On the topology defined by Thurston's asymmetric metric, Math. Proc. Cambridge Philos. Soc., Volume 142 (2007), pp. 487-496 (ISSN: 0305-0041) | DOI | MR | Zbl

Rivin, I.; Schlenker, J.-M. The Schläfli formula in Einstein manifolds with boundary, Electron. Res. Announc. Amer. Math. Soc., Volume 5 (1999), pp. 18-23 (ISSN: 1079-6762) | DOI | MR | Zbl

Sampson, J. H. Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup., Volume 11 (1978), pp. 211-228 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Sözen, Y.; Bonahon, F. The Weil-Petersson and Thurston symplectic forms, Duke Math. J., Volume 108 (2001), pp. 581-597 (ISSN: 0012-7094) | DOI | MR | Zbl

Schlenker, J.-M. Hyperbolic manifolds with convex boundary, Invent. Math., Volume 163 (2006), pp. 109-169 (ISSN: 0020-9910) | DOI | MR | Zbl

Schoen, R. M., Complex geometry (Osaka, 1990) (Lecture Notes in Pure and Appl. Math.), Volume 143, Dekker, New York, 1993, pp. 179-200 | MR | Zbl

Schlenker, J.-M. Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom., Volume 4 (1996), pp. 285-331 (ISSN: 1019-8385) | DOI | MR | Zbl

Strebel, K., Ergebn. Math. Grenzg., 5, Springer, Berlin, 1984, 184 pages (ISBN: 3-540-13035-7) | DOI | MR | Zbl

Scannell, K. P.; Wolf, M. The grafting map of Teichmüller space, J. Amer. Math. Soc., Volume 15 (2002), pp. 893-927 (ISSN: 0894-0347) | DOI | MR | Zbl

Thurston, W. P. Minimal stretch maps between hyperbolic surfaces (preprint arXiv:math/9801039 ) | MR

Thurston, W. P., Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) (London Math. Soc. Lecture Note Ser.), Volume 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91-112 | MR | Zbl

Tromba, A. J. On an energy function for the Weil-Petersson metric on Teichmüller space, Manuscripta Math., Volume 59 (1987), pp. 249-260 (ISSN: 0025-2611) | DOI | MR | Zbl

Tromba, A. J., Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 315-341 | MR | Zbl

Tromba, A. J. Dirichlet's energy on Teichmüller's moduli space and the Nielsen realization problem, Math. Z., Volume 222 (1996), pp. 451-464 (ISSN: 0025-5874) | DOI | MR | Zbl

Takhtajan, L. A.; Teo, L.-P. Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys., Volume 239 (2003), pp. 183-240 (ISSN: 0010-3616) | DOI | MR | Zbl

Takhtajan, L. A.; Zograf, P. G. Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on $ℳ_{0, n}$ , Trans. Amer. Math. Soc., Volume 355 (2003), pp. 1857-1867 (ISSN: 0002-9947) | DOI | MR | Zbl

Wolf, M. The Weil-Petersson Hessian of length on Teichmüller space, J. Differential Geom., Volume 91 (2012), pp. 129-169 http://projecteuclid.org/euclid.jdg/1343133703 (ISSN: 0022-040X) | MR | Zbl

Wolpert, S. A. On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math., Volume 117 (1983), pp. 207-234 (ISSN: 0003-486X) | DOI | MR | Zbl

Wolpert, S. A. Geodesic length functions and the Nielsen problem, J. Differential Geom., Volume 25 (1987), pp. 275-296 http://projecteuclid.org/euclid.jdg/1214440853 (ISSN: 0022-040X) | MR | Zbl

Wolf, M. The Teichmüller theory of harmonic maps, J. Differential Geom., Volume 29 (1989), pp. 449-479 http://projecteuclid.org/euclid.jdg/1214442885 (ISSN: 0022-040X) | MR | Zbl

Wolf, M. Harmonic maps from surfaces to $𝐑$ -trees, Math. Z., Volume 218 (1995), pp. 577-593 (ISSN: 0025-5874) | DOI | MR | Zbl

Zograf, P. G.; Takhtadzhyan, L. A. On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces, Mat. Sb. (N.S.), Volume 132 (1987), pp. 304-321 ; English translation: Math. USSR Sb. 60 (1988), 297–313 (ISSN: 0368-8666) | MR | Zbl

Cité par Sources :