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.

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: 10.5802/aif.2622
Classification: 53C50, 53C42, 53C80
Keywords: Gauss curvature, $K$-curvature, Minkowski problem
Mot clés : courbure de Gauss, $K$-courbure, problème de Minkowski
Barbot, Thierry 1; Béguin, François 2; Zeghib, Abdelghani 3

1 Université d’Avignon et des pays de Vaucluse Faculté des Sciences Laboratoire d’Analyse non Linéaire et Géométrie 33 rue Louis Pasteur 84000 Avignon (France)
2 Université Paris Sud Laboratoire de Mathématiques Bâtiment 425 91425 Orsay Cedex (France)
3 École Normale Supérieure de Lyon CNRS, UMPA 46, allée d’Italie 69364 LYON Cedex 07(France)
@article{AIF_2011__61_2_511_0,
     author = {Barbot, Thierry and B\'eguin, Fran\c{c}ois 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{\textquoteright}institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2622},
     zbl = {1234.53019},
     mrnumber = {2895066},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2622/}
}
TY  - JOUR
AU  - Barbot, Thierry
AU  - Béguin, François
AU  - Zeghib, Abdelghani
TI  - Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 511
EP  - 591
VL  - 61
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2622/
DO  - 10.5802/aif.2622
LA  - en
ID  - AIF_2011__61_2_511_0
ER  - 
%0 Journal Article
%A Barbot, Thierry
%A Béguin, François
%A Zeghib, Abdelghani
%T Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space
%J Annales de l'Institut Fourier
%D 2011
%P 511-591
%V 61
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2622/
%R 10.5802/aif.2622
%G en
%F 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, Volume 61 (2011) no. 2, pp. 511-591. doi : 10.5802/aif.2622. http://www.numdam.org/articles/10.5802/aif.2622/

[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 | Zbl

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR

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

[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 | Zbl

[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 | Zbl

[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 | DOI | MR

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

[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., Volume 69 (2005) no. 3, pp. 441-521 | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[42] Iskhakov, I. On hyperbolic surface tessellations and equivariant spacelike polyhedral surfaces in Minkowski space, Ohio State University (2000) (Ph. D. Thesis www.math.ohio-state.edu/~mdavis/eprints.html)

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

Cited by Sources: