Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Thierry Barbot[1]; François Béguin[2]; Abdelghani Zeghib[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)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 2, page 511-591
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Barbot, Thierry, Béguin, François, and 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 61.2 (2011): 511-591. <http://eudml.org/doc/219723>.

@article{Barbot2011,
abstract = {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 $\mathrm\{Min\}_\{3\}$ for data that are invariant under the action of a co-compact Fuchsian group.},
affiliation = {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); Université Paris Sud Laboratoire de Mathématiques Bâtiment 425 91425 Orsay Cedex (France); École Normale Supérieure de Lyon CNRS, UMPA 46, allée d’Italie 69364 LYON Cedex 07(France)},
author = {Barbot, Thierry, Béguin, François, Zeghib, Abdelghani},
journal = {Annales de l’institut Fourier},
keywords = {Gauss curvature; $K$-curvature; Minkowski problem; -curvature},
language = {eng},
number = {2},
pages = {511-591},
publisher = {Association des Annales de l’institut Fourier},
title = {Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space},
url = {http://eudml.org/doc/219723},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 511
EP - 591
AB - 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 $\mathrm{Min}_{3}$ for data that are invariant under the action of a co-compact Fuchsian group.
LA - eng
KW - Gauss curvature; $K$-curvature; Minkowski problem; -curvature
UR - http://eudml.org/doc/219723
ER -

References

top
  1. L. Andersson, T. Barbot, F. Béguin, A. Zeghib, Cosmological time versus CMC time I: Flat spacetimes Zbl1246.53093
  2. L. Andersson, T. Barbot, F. Béguin, A. Zeghib, Cosmological time versus CMC time II: the de Sitter and anti-de Sitter cases Zbl1246.53093
  3. Lars Andersson, Constant mean curvature foliations of flat space-times, Comm. Anal. Geom. 10 (2002), 1125-1150 Zbl1038.53025MR1957665
  4. Lars Andersson, Constant mean curvature foliations of simplicial flat spacetimes, Comm. Anal. Geom. 13 (2005), 963-979 Zbl1123.53034MR2216148
  5. Lars Andersson, Gregory J. Galloway, Ralph Howard, The cosmological time function, Classical Quantum Gravity 15 (1998), 309-322 Zbl0911.53039MR1606594
  6. Lars Andersson, Vincent Moncrief, Future complete vacuum spacetimes, The Einstein equations and the large scale behavior of gravitational fields (2004), 299-330, Birkhäuser, Basel Zbl1105.83001MR2098919
  7. Lars Andersson, Vincent Moncrief, Anthony J. Tromba, On the global evolution problem in 2 + 1 gravity, J. Geom. Phys. 23 (1997), 191-205 Zbl0898.58003MR1484587
  8. Máximo Bañados, Marc Henneaux, Claudio Teitelboim, Jorge Zanelli, Geometry of the 2 + 1 black hole, Phys. Rev. D (3) 48 (1993), 1506-1525 MR1236812
  9. Thierry Barbot, Globally hyperbolic flat space-times, J. Geom. Phys. 53 (2005), 123-165 Zbl1087.53065MR2110829
  10. Thierry Barbot, Causal properties of AdS-isometry groups. II. BTZ multi-black-holes, Adv. Theor. Math. Phys. 12 (2008), 1209-1257 Zbl1153.83349MR2443264
  11. Thierry Barbot, François Béguin, Abdelghani Zeghib, Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante, C. R. Math. Acad. Sci. Paris 336 (2003), 245-250 Zbl1026.53015MR1968267
  12. Thierry Barbot, François Béguin, Abdelghani Zeghib, Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on A d S 3 , Geom. Dedicata 126 (2007), 71-129 Zbl1255.83118MR2328923
  13. Thierry Barbot, Abdelghani Zeghib, Group actions on Lorentz spaces, mathematical aspects: a survey, The Einstein equations and the large scale behavior of gravitational fields (2004), 401-439, Birkhäuser, Basel Zbl1064.53049MR2098923
  14. Robert Bartnik, Leon Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), 131-152 Zbl0512.53055MR680653
  15. Pierre Bayard, Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in ℝ n , 1 , Calc. Var. Partial Differential Equations 18 (2003), 1-30 Zbl1043.53027MR2001880
  16. Pierre Bayard, Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space, Calc. Var. Partial Differential Equations 26 (2006), 245-264 Zbl1114.35069MR2222246
  17. John K. Beem, Paul E. Ehrlich, Kevin L. Easley, Global Lorentzian geometry, 202 (1996), Marcel Dekker Inc., New York Zbl0462.53001MR1384756
  18. Riccardo Benedetti, Francesco Bonsante, Canonical Wick rotations in 3-dimensional gravity, Mem. Amer. Math. Soc. 198 (2009) Zbl1165.53047MR2499272
  19. Riccardo Benedetti, Enore Guadagnini, Cosmological time in ( 2 + 1 ) -gravity, Nuclear Phys. B 613 (2001), 330-352 Zbl0970.83039MR1857817
  20. M. S. Berger, Riemannian structure of prescribed Gauss curvature for 2-manifolds, J. Diff. Geom. 5 (1971), 325-332 Zbl0222.53042MR295261
  21. Francis Bonahon, Geodesic laminations with transverse Hölder distributions, Ann. Sci. École Norm. Sup. (4) 30 (1997), 205-240 Zbl0887.57018MR1432054
  22. Francesco Bonsante, Deforming the Minkowskian cone of a closed hyperbolic manifold, (2005) Zbl1094.53063
  23. Francesco Bonsante, Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differential Geom. 69 (2005), 441-521 Zbl1094.53063MR2170277
  24. Peter Buser, Geometry and spectra of compact Riemann surfaces, 106 (1992), Birkhäuser Boston Inc., Boston, MA Zbl1239.32001MR1183224
  25. Steven Carlip, Quantum gravity in 2 + 1 dimensions, (1998), Cambridge University Press, Cambridge Zbl0919.53024MR1637718
  26. Shiu Yuen Cheng, Shing Tung Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), 407-419 Zbl0352.53021MR431061
  27. Shiu Yuen Cheng, Shing Tung Yau, On the regularity of the solution of the n -dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), 495-516 Zbl0363.53030MR423267
  28. H. S. M. Coxeter, A geometrical background for de Sitter’s world, Amer. Math. Monthly 50 (1943), 217-228 Zbl0060.44309MR7991
  29. Gaston Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, (1887-1896), Gauthier-Villars Zbl0257.53001
  30. F. Delanöe, The Dirichlet problem for an equation of given Lorentz-Gaussian curvature, Ukrain. Mat. Zh. 42 (1990), 1704-1710 Zbl0724.35039MR1098472
  31. Klaus Ecker, Gerhard Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), 595-613 Zbl0721.53055MR1091580
  32. J.-H. Eschenburg, G. J. Galloway, Lines in space-times, Comm. Math. Phys. 148 (1992), 209-216 Zbl0756.53028MR1178143
  33. Claus Gerhardt, Minkowki type problems for convex hypersurfaces in hyperbolic space Zbl0932.35090
  34. Claus Gerhardt, Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana Univ. Math. J. 49 (2000), 1125-1153 Zbl1034.53064MR1803223
  35. Claus Gerhardt, Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. Reine Angew. Math. 554 (2003), 157-199 Zbl1091.53039MR1952172
  36. Claus Gerhardt, Minkowski type problems for convex hypersurfaces in the sphere, Pure Appl. Math. Q. 3 (2007), 417-449 Zbl1152.53043MR2340049
  37. Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437-449 Zbl0189.27602MR270697
  38. Bo Guan, The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc. 350 (1998), 4955-4971 Zbl0919.35046MR1451602
  39. Bo Guan, Pengfei Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2) 156 (2002), 655-673 Zbl1025.53028MR1933079
  40. Bo Guan, Huai-Yu Jian, Richard M. Schoen, Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space, J. Reine Angew. Math. 595 (2006), 167-188 Zbl1097.53040MR2244801
  41. Jun-ichi Hano, Katsumi Nomizu, On isometric immersions of the hyperbolic plane into the Lorentz-Minkowski space and the Monge-Ampère equation of a certain type, Math. Ann. 262 (1983), 245-253 Zbl0507.53042MR690199
  42. I. Iskhakov, On hyperbolic surface tessellations and equivariant spacelike polyhedral surfaces in Minkowski space, (2000) 
  43. Kirill Krasnov, Jean-Marc Schlenker, Minimal surfaces and particles in 3-manifolds, Geom. Dedicata 126 (2007), 187-254 Zbl1126.53037MR2328927
  44. François Labourie, Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France 119 (1991), 307-325 Zbl0758.53030MR1125669
  45. François Labourie, Problèmes de Monge-Ampère, courbes holomorphes et laminations, Geom. Funct. Anal. 7 (1997), 496-534 Zbl0885.32013MR1466336
  46. Gilbert Levitt, Foliations and laminations on hyperbolic surfaces, Topology 22 (1983), 119-135 Zbl0522.57027MR683752
  47. An Min Li, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space, Arch. Math. (Basel) 64 (1995), 534-551 Zbl0828.53050MR1329827
  48. An Min Li, Udo Simon, Guo Song Zhao, Global affine differential geometry of hypersurfaces, 11 (1993), Walter de Gruyter & Co., Berlin Zbl0808.53002MR1257186
  49. R. Mazzeo, F. Pacard, Constant curvature foliations in asymptotically hyperbolic spaces Zbl1214.53024
  50. William H. Meeks, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), 539-552 Zbl0617.53007MR940118
  51. Geoffrey Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007), 3-45 Zbl1206.83117MR2328921
  52. Vincent Moncrief, Reduction of the Einstein equations in 2 + 1 dimensions to a Hamiltonian system over Teichmüller space, J. Math. Phys. 30 (1989), 2907-2914 Zbl0704.53076MR1025234
  53. Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394 Zbl0051.12402MR58265
  54. Barrett O’Neill, Semi-Riemannian geometry, 103 (1983), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York Zbl0531.53051MR719023
  55. Kevin P. Scannell, Flat conformal structures and the classification of de Sitter manifolds, Comm. Anal. Geom. 7 (1999), 325-345 Zbl0941.53040MR1685590
  56. Jean-Marc Schlenker, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom. 4 (1996), 285-331 Zbl0864.53016MR1393565
  57. Oliver C. Schnürer, The Dirichlet problem for Weingarten hypersurfaces in Lorentz manifolds, Math. Z. 242 (2002), 159-181 Zbl1042.53026MR1985454
  58. Oliver C. Schnürer, A generalized Minkowski problem with Dirichlet boundary condition, Trans. Amer. Math. Soc. 355 (2003), 655-663 (electronic) Zbl1081.35045MR1932719
  59. G. Smith, Moduli of flat conformal structures of hyperbolic type Zbl1231.53012
  60. Andrejs E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, V, Invent. Math. 66 (1982), 39-56 Zbl0483.53055MR652645
  61. John Urbas, The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space, Calc. Var. Partial Differential Equations 18 (2003), 307-316 Zbl1080.53062MR2018670

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.