Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, p. 511-591
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.
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.
DOI : https://doi.org/10.5802/aif.2622
Classification:  53C50,  53C42,  53C80
Keywords: Gauss curvature, K-curvature, Minkowski problem
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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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     pages = {511-591},
     doi = {10.5802/aif.2622},
     zbl = {1234.53019},
     mrnumber = {2895066},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_2_511_0}
}
Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space. Annales de l'Institut Fourier, Volume 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., Tome 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., Tome 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, Tome 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., Tome 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), Tome 48 (1993) no. 4, pp. 1506-1525 | Article | MR 1236812

[9] Barbot, Thierry Globally hyperbolic flat space-times, J. Geom. Phys., Tome 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., Tome 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, Tome 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, Tome 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., Tome 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, Tome 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, Tome 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, Marcel Dekker Inc., New York, Monographs and Textbooks in Pure and Applied Mathematics, Tome 202 (1996) | MR 1384756 | Zbl 0462.53001

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

[19] Benedetti, Riccardo; Guadagnini, Enore Cosmological time in (2+1)-gravity, Nuclear Phys. B, Tome 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., Tome 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), Tome 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, Pisa (2005) (Ph. D. Thesis)

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

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

[25] Carlip, Steven Quantum gravity in 2 + 1 dimensions, Cambridge University Press, Cambridge, Cambridge Monographs on Mathematical Physics (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), Tome 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., Tome 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, Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 3 (2007) no. 2, part 1, pp. 417-449 | MR 2340049 | Zbl 1152.53043

[37] Geroch, Robert Domain of dependence, J. Mathematical Phys., Tome 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., Tome 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), Tome 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., Tome 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., Tome 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, Ohio State University (2000) (Ph. D. Thesis)

[43] Krasnov, Kirill; Schlenker, Jean-Marc Minimal surfaces and particles in 3-manifolds, Geom. Dedicata, Tome 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, Tome 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., Tome 7 (1997) no. 3, pp. 496-534 | Article | MR 1466336 | Zbl 0885.32013

[46] Levitt, Gilbert Foliations and laminations on hyperbolic surfaces, Topology, Tome 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), Tome 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, Walter de Gruyter & Co., Berlin, de Gruyter Expositions in Mathematics, Tome 11 (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., Tome 27 (1988) no. 3, pp. 539-552 | MR 940118 | Zbl 0617.53007

[51] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Tome 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., Tome 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., Tome 6 (1953), pp. 337-394 | Article | MR 58265 | Zbl 0051.12402

[54] O’Neill, Barrett Semi-Riemannian geometry, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, Pure and Applied Mathematics, Tome 103 (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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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, Tome 18 (2003) no. 3, pp. 307-316 | Article | MR 2018670 | Zbl 1080.53062