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

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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

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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 -

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