Maximal surfaces in anti-de Sitter 3-manifolds with particles
[Surfaces maximales dans les variétés anti-de Sitter de dimension 3 à particules]
Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1409-1449.

On démontre l’existence d’une unique surface maximale dans les variétés anti-de Sitter (AdS) globalement hyperboliques maximales (GHM) à particules (c’est à dire, avec des singularités coniques le long de courbes de type temps) lorsque les angles sont inférieurs à π. On interprète ce résultat en termes de théorie de Teichmüller et nous démontrons l’existence d’un unique difféomorphisme minimal lagrangien isotope à l’identité entre deux surfaces hyperboliques à singularités coniques, lorsque les angles singuliers sont les mêmes pour les deux surfaces et sont inférieurs à π.

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than π. We interpret this result in terms of Teichmüller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than π.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3040
Classification : 53C42, 53C50
Keywords: maximal surfaces, cone-manifolds, Lorentz geometry, minimal Lagrangian maps
Mot clés : surfaces maximales, variétés à singularités coniques, géometrie lorentzienne, applications minimales lagrangiennes
Toulisse, Jérémy 1

1 Department of Mathematics Mathematics Research Unit BLG University of Southern Califonia 3620 S. Vermont Avenue, KAP 104 Los Angeles, CA 90089-2532 (USA)
@article{AIF_2016__66_4_1409_0,
     author = {Toulisse, J\'er\'emy},
     title = {Maximal surfaces in anti-de {Sitter} 3-manifolds with particles},
     journal = {Annales de l'Institut Fourier},
     pages = {1409--1449},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     doi = {10.5802/aif.3040},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3040/}
}
TY  - JOUR
AU  - Toulisse, Jérémy
TI  - Maximal surfaces in anti-de Sitter 3-manifolds with particles
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1409
EP  - 1449
VL  - 66
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3040/
DO  - 10.5802/aif.3040
LA  - en
ID  - AIF_2016__66_4_1409_0
ER  - 
%0 Journal Article
%A Toulisse, Jérémy
%T Maximal surfaces in anti-de Sitter 3-manifolds with particles
%J Annales de l'Institut Fourier
%D 2016
%P 1409-1449
%V 66
%N 4
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3040/
%R 10.5802/aif.3040
%G en
%F AIF_2016__66_4_1409_0
Toulisse, Jérémy. Maximal surfaces in anti-de Sitter 3-manifolds with particles. Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1409-1449. doi : 10.5802/aif.3040. http://archive.numdam.org/articles/10.5802/aif.3040/

[1] Aiyama, R.; Akutagawa, K.; Wan, T. Y. H. Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space, Tohoku Math. J. (2), Volume 52 (2000) no. 3, pp. 415-429 | DOI

[2] Andersson, L.; Barbot, T.; Béguin, F.; Zeghib, A. Cosmological time versus CMC time in spacetimes of constant curvature, Asian J. Math., Volume 16 (2012) no. 1, pp. 37-87 | DOI

[3] 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 (2012) (http://arxiv.org/abs/1205.6103)

[4] Barbot, Thierry; Béguin, François; Zeghib, Abdelghani Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS 3 , Geom. Dedicata, Volume 126 (2007), pp. 71-129 | DOI

[5] Bers, L. Simultaneous uniformization, Bull. Amer. Math. Soc., Volume 66 (1960), pp. 94-97

[6] Bonsante, F.; Schlenker, J.-M. AdS manifolds with particles and earthquakes on singular surfaces, Geom. Funct. Anal., Volume 19 (2009) no. 1, pp. 41-82 | DOI

[7] Choquet-Bruhat, Yvonne; Geroch, Robert Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., Volume 14 (1969), pp. 329-335

[8] Eells, J. J.; Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160

[9] Gell-Redman, J. Harmonic maps of conic surfaces with cone angles less than 2π, Comm. Anal. Geom., Volume 23 (2015) no. 4, pp. 717-796 | DOI

[10] Gerhardt, C. H-surfaces in Lorentzian manifolds, Comm. Math. Phys., Volume 89 (1983) no. 4, pp. 523-553 http://projecteuclid.org/getRecord?id=euclid.cmp/1103922929

[11] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, xiv+517 pages (Reprint of the 1998 edition)

[12] Goldman, William M. Topological components of spaces of representations, Invent. Math., Volume 93 (1988) no. 3, pp. 557-607 | DOI

[13] Hopf, H. Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr., Volume 4 (1951), pp. 232-249

[14] Jeffres, T. D; Mazzeo, R.; Rubinstein, Y. A Kähler-Einstein metrics with edge singularities (2011) (http://arxiv.org/abs/1105.5216)

[15] Krasnov, K.; Schlenker, J.-M. Minimal surfaces and particles in 3-manifolds, Geom. Dedicata, Volume 126 (2007), pp. 187-254 | DOI

[16] Labourie, F. Surfaces convexes dans l’espace hyperbolique et ℂℙ 1 -structures, J. London Math. Soc. (2), Volume 45 (1992) no. 3, pp. 549-565 | DOI

[17] Lecuire, C.; Schlenker, J.-M. The convex core of quasifuchsian manifolds with particles, Geom. Topol., Volume 18 (2014) no. 4, pp. 2309-2373 | DOI

[18] Mazzeo, R.; Rubinstein, Y. A.; Sesum, N. Ricci flow on surfaces with conic singularities, Anal. PDE, Volume 8 (2015) no. 4, pp. 839-882 | DOI

[19] McOwen, R. C. Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc., Volume 103 (1988) no. 1, pp. 222-224 | DOI

[20] Mess, G. Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | DOI

[21] Moroianu, S.; Schlenker, J.-M. Quasi-Fuchsian manifolds with particles, J. Differential Geom., Volume 83 (2009) no. 1, pp. 75-129 http://projecteuclid.org/getRecord?id=euclid.jdg/1253804352

[22] Schlenker, J.-M. Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom., Volume 48 (1998) no. 2, pp. 323-405 http://projecteuclid.org/getRecord?id=euclid.jdg/1214460799

[23] Schoen, Richard M. The role of harmonic mappings in rigidity and deformation problems, Complex geometry (Osaka, 1990) (Lecture Notes in Pure and Appl. Math.), Volume 143, Dekker, New York, 1993, pp. 179-200

[24] Toulisse, J. Minimal diffeomorphism between hyperbolic surfaces with cone singularities (2014) (http://arxiv.org/abs/1411.2656v1)

[25] Troyanov, M. Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., Volume 324 (1991) no. 2, pp. 793-821 | DOI

Cité par Sources :