Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space  [ Surfaces à courbure de Gauss prescrite dans les espaces-temps de dimension 3 - Application au problème de Minkowski dans l’espace de Minkowski ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 511-591.

Nous étudions l’existence de surfaces à courbure de Gauss constante ou prescrite dans certains espaces-temps lorentziens. Nous montrons en particulier que tout espace-temps (non-élémentaire) globalement hyperbolique spatialement compact maximal à courbure constante positive ou nulle de dimension 3 est feuilleté en surfaces de Cauchy à courbure de Gauss constante. Dans le cas des espaces-temps à courbure constante strictement négative, le complémentaire du cœur convexe est feuilleté par des surfaces de Cauchy à courbure de Gauss constante. On combinant ces résultats d’existence de feuilletages avec un théorème de C. Gerhardt, on obtient un certain nombre de corollaires. Par exemple, on résout le problème de Minkowski dans Min 3 pour des données qui sont invariantes par l’action d’un groupe fuchsien cocompact.

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in Min 3 for data that are invariant under the action of a co-compact Fuchsian group.

Reçu le : 2008-02-05
Révisé le : 2009-11-04
Accepté le : 2010-01-04
DOI : https://doi.org/10.5802/aif.2622
Classification : 53C50,  53C42,  53C80
Mots clés: courbure de Gauss, K-courbure, problème de Minkowski
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     author = {Barbot, Thierry and B\'eguin, Fran\c cois and Zeghib, Abdelghani},
     title = {Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space},
     journal = {Annales de l'Institut Fourier},
     pages = {511--591},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2622},
     zbl = {1234.53019},
     mrnumber = {2895066},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2011__61_2_511_0/}
}
Barbot, Thierry; Béguin, François; Zeghib, Abdelghani. Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 511-591. doi : 10.5802/aif.2622. https://aif.centre-mersenne.org/item/AIF_2011__61_2_511_0/

[1] Andersson, L.; Barbot, T.; Béguin, F.; Zeghib, A. Cosmological time versus CMC time I: Flat spacetimes (arXiv: math.DG/0604486)

[2] Andersson, L.; Barbot, T.; Béguin, F.; Zeghib, A. Cosmological time versus CMC time II: the de Sitter and anti-de Sitter cases (arXiv: math.DG/0701452)

[3] Andersson, Lars Constant mean curvature foliations of flat space-times, Comm. Anal. Geom., Volume 10 (2002) no. 5, pp. 1125-1150 | MR 1957665 | Zbl 1038.53025

[4] Andersson, Lars Constant mean curvature foliations of simplicial flat spacetimes, Comm. Anal. Geom., Volume 13 (2005) no. 5, pp. 963-979 | MR 2216148 | Zbl 1123.53034

[5] Andersson, Lars; Galloway, Gregory J.; Howard, Ralph The cosmological time function, Classical Quantum Gravity, Volume 15 (1998) no. 2, pp. 309-322 | Article | MR 1606594 | Zbl 0911.53039

[6] Andersson, Lars; Moncrief, Vincent Future complete vacuum spacetimes, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 299-330 | MR 2098919 | Zbl 1105.83001

[7] Andersson, Lars; Moncrief, Vincent; Tromba, Anthony J. On the global evolution problem in 2+1 gravity, J. Geom. Phys., Volume 23 (1997) no. 3-4, pp. 191-205 | Article | MR 1484587 | Zbl 0898.58003

[8] Bañados, Máximo; Henneaux, Marc; Teitelboim, Claudio; Zanelli, Jorge Geometry of the 2+1 black hole, Phys. Rev. D (3), Volume 48 (1993) no. 4, pp. 1506-1525 | Article | MR 1236812

[9] Barbot, Thierry Globally hyperbolic flat space-times, J. Geom. Phys., Volume 53 (2005) no. 2, pp. 123-165 | Article | MR 2110829 | Zbl 1087.53065

[10] Barbot, Thierry Causal properties of AdS-isometry groups. II. BTZ multi-black-holes, Adv. Theor. Math. Phys., Volume 12 (2008) no. 6, pp. 1209-1257 | MR 2443264 | Zbl 1153.83349

[11] Barbot, Thierry; Béguin, François; Zeghib, Abdelghani Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 3, pp. 245-250 | MR 1968267 | Zbl 1026.53015

[12] Barbot, Thierry; Béguin, François; Zeghib, Abdelghani Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS 3 , Geom. Dedicata, Volume 126 (2007) no. 1, pp. 71-129 | Article | MR 2328923 | Zbl pre05200417

[13] Barbot, Thierry; Zeghib, Abdelghani Group actions on Lorentz spaces, mathematical aspects: a survey, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 401-439 | MR 2098923 | Zbl 1064.53049

[14] Bartnik, Robert; Simon, Leon Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., Volume 87 (1982/83) no. 1, pp. 131-152 | Article | MR 680653 | Zbl 0512.53055

[15] Bayard, Pierre Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in n,1 , Calc. Var. Partial Differential Equations, Volume 18 (2003) no. 1, pp. 1-30 | Article | MR 2001880 | Zbl 1043.53027

[16] Bayard, Pierre Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space, Calc. Var. Partial Differential Equations, Volume 26 (2006) no. 2, pp. 245-264 | Article | MR 2222246 | Zbl 1114.35069

[17] Beem, John K.; Ehrlich, Paul E.; Easley, Kevin L. Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Mathematics, Volume 202, Marcel Dekker Inc., New York, 1996 | MR 1384756 | Zbl 0462.53001

[18] Benedetti, Riccardo; Bonsante, Francesco Canonical Wick rotations in 3-dimensional gravity, Mem. Amer. Math. Soc., Volume 198 (2009) no. 926, viii+164 pages | MR 2499272 | Zbl 1165.53047

[19] Benedetti, Riccardo; Guadagnini, Enore Cosmological time in (2+1)-gravity, Nuclear Phys. B, Volume 613 (2001) no. 1-2, pp. 330-352 | Article | MR 1857817 | Zbl 0970.83039

[20] Berger, M. S. Riemannian structure of prescribed Gauss curvature for 2-manifolds, J. Diff. Geom., Volume 5 (1971), pp. 325-332 | MR 295261 | Zbl 0222.53042

[21] Bonahon, Francis Geodesic laminations with transverse Hölder distributions, Ann. Sci. École Norm. Sup. (4), Volume 30 (1997) no. 2, pp. 205-240 | Numdam | MR 1432054 | Zbl 0887.57018

[22] Bonsante, Francesco Deforming the Minkowskian cone of a closed hyperbolic manifold (2005) (Ph. D. Thesis)

[23] Bonsante, Francesco Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differential Geom., Volume 69 (2005) no. 3, pp. 441-521 | MR 2170277 | Zbl 1094.53063

[24] Buser, Peter Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, Volume 106, Birkhäuser Boston Inc., Boston, MA, 1992 | MR 1183224 | Zbl pre05814173

[25] Carlip, Steven Quantum gravity in 2+1 dimensions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1998 | MR 1637718 | Zbl 0919.53024

[26] Cheng, Shiu Yuen; Yau, Shing Tung Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2), Volume 104 (1976) no. 3, pp. 407-419 | Article | MR 431061 | Zbl 0352.53021

[27] Cheng, Shiu Yuen; Yau, Shing Tung On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math., Volume 29 (1976) no. 5, pp. 495-516 | Article | MR 423267 | Zbl 0363.53030

[28] Coxeter, H. S. M. A geometrical background for de Sitter’s world, Amer. Math. Monthly, Volume 50 (1943), pp. 217-228 | Article | MR 7991 | Zbl 0060.44309

[29] Darboux, Gaston Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Gauthier-Villars, 1887-1896

[30] Delanöe, F. The Dirichlet problem for an equation of given Lorentz-Gaussian curvature, Ukrain. Mat. Zh., Volume 42 (1990) no. 12, pp. 1704-1710 | MR 1098472 | Zbl 0724.35039

[31] Ecker, Klaus; Huisken, Gerhard Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., Volume 135 (1991) no. 3, pp. 595-613 | Article | MR 1091580 | Zbl 0721.53055

[32] Eschenburg, J.-H.; Galloway, G. J. Lines in space-times, Comm. Math. Phys., Volume 148 (1992) no. 1, pp. 209-216 | Article | MR 1178143 | Zbl 0756.53028

[33] Gerhardt, Claus Minkowki type problems for convex hypersurfaces in hyperbolic space (arXiv: math.DG/0602597)

[34] Gerhardt, Claus Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana Univ. Math. J., Volume 49 (2000) no. 3, pp. 1125-1153 | Article | MR 1803223 | Zbl 1034.53064

[35] Gerhardt, Claus Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. Reine Angew. Math., Volume 554 (2003), pp. 157-199 | Article | MR 1952172 | Zbl 1091.53039

[36] Gerhardt, Claus Minkowski type problems for convex hypersurfaces in the sphere, Pure Appl. Math. Q., Volume 3 (2007) no. 2, part 1, pp. 417-449 | MR 2340049 | Zbl 1152.53043

[37] Geroch, Robert Domain of dependence, J. Mathematical Phys., Volume 11 (1970), pp. 437-449 | Article | MR 270697 | Zbl 0189.27602

[38] Guan, Bo The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc., Volume 350 (1998) no. 12, pp. 4955-4971 | Article | MR 1451602 | Zbl 0919.35046

[39] Guan, Bo; Guan, Pengfei Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2), Volume 156 (2002) no. 2, pp. 655-673 | Article | MR 1933079 | Zbl 1025.53028

[40] Guan, Bo; Jian, Huai-Yu; Schoen, Richard M. Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space, J. Reine Angew. Math., Volume 595 (2006), pp. 167-188 | Article | MR 2244801 | Zbl 1097.53040

[41] Hano, Jun-ichi; Nomizu, Katsumi On isometric immersions of the hyperbolic plane into the Lorentz-Minkowski space and the Monge-Ampère equation of a certain type, Math. Ann., Volume 262 (1983) no. 2, pp. 245-253 | Article | MR 690199 | Zbl 0507.53042

[42] Iskhakov, I. On hyperbolic surface tessellations and equivariant spacelike polyhedral surfaces in Minkowski space (2000) (Ph. D. Thesis)

[43] Krasnov, Kirill; Schlenker, Jean-Marc Minimal surfaces and particles in 3-manifolds, Geom. Dedicata, Volume 126 (2007), pp. 187-254 | Article | MR 2328927 | Zbl 1126.53037

[44] Labourie, François Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France, Volume 119 (1991) no. 3, pp. 307-325 | Numdam | MR 1125669 | Zbl 0758.53030

[45] Labourie, François Problèmes de Monge-Ampère, courbes holomorphes et laminations, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 496-534 | Article | MR 1466336 | Zbl 0885.32013

[46] Levitt, Gilbert Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983) no. 2, pp. 119-135 | Article | MR 683752 | Zbl 0522.57027

[47] Li, An Min Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space, Arch. Math. (Basel), Volume 64 (1995) no. 6, pp. 534-551 | MR 1329827 | Zbl 0828.53050

[48] Li, An Min; Simon, Udo; Zhao, Guo Song Global affine differential geometry of hypersurfaces, de Gruyter Expositions in Mathematics, Volume 11, Walter de Gruyter & Co., Berlin, 1993 | MR 1257186 | Zbl 0808.53002

[49] Mazzeo, R.; Pacard, F. Constant curvature foliations in asymptotically hyperbolic spaces (arXiv: 0710.2298)

[50] Meeks, William H. The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom., Volume 27 (1988) no. 3, pp. 539-552 | MR 940118 | Zbl 0617.53007

[51] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | Article | MR 2328921 | Zbl 1206.83117

[52] Moncrief, Vincent Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space, J. Math. Phys., Volume 30 (1989) no. 12, pp. 2907-2914 | Article | MR 1025234 | Zbl 0704.53076

[53] Nirenberg, Louis The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., Volume 6 (1953), pp. 337-394 | Article | MR 58265 | Zbl 0051.12402

[54] O’Neill, Barrett Semi-Riemannian geometry, Pure and Applied Mathematics, Volume 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983 (With applications to relativity) | MR 719023 | Zbl 0531.53051

[55] Scannell, Kevin P. Flat conformal structures and the classification of de Sitter manifolds, Comm. Anal. Geom., Volume 7 (1999) no. 2, pp. 325-345 | MR 1685590 | Zbl 0941.53040

[56] Schlenker, Jean-Marc Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom., Volume 4 (1996) no. 1-2, pp. 285-331 | MR 1393565 | Zbl 0864.53016

[57] Schnürer, Oliver C. The Dirichlet problem for Weingarten hypersurfaces in Lorentz manifolds, Math. Z., Volume 242 (2002) no. 1, pp. 159-181 | Article | MR 1985454 | Zbl 1042.53026

[58] Schnürer, Oliver C. A generalized Minkowski problem with Dirichlet boundary condition, Trans. Amer. Math. Soc., Volume 355 (2003) no. 2, p. 655-663 (electronic) | Article | MR 1932719 | Zbl 1081.35045

[59] Smith, G. Moduli of flat conformal structures of hyperbolic type (arXiv:0804.0744)

[60] Treibergs, Andrejs E. Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, V, Invent. Math., Volume 66 (1982) no. 1, pp. 39-56 | Article | MR 652645 | Zbl 0483.53055

[61] Urbas, John The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space, Calc. Var. Partial Differential Equations, Volume 18 (2003) no. 3, pp. 307-316 | Article | MR 2018670 | Zbl 1080.53062