Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Abstract Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.

[1]  L. Alías,et al.  SPACELIKE HYPERSURFACES OF CONSTANT MEAN CURVATURE AND CALABI-BERNSTEIN TYPE PROBLEMS , 1997 .

[2]  Osamu Kobayashi.  Maximal Surfaces in the 3-Dimensional Minkowski Space $L^3$ , 1983 .

[3]  Alma L. Albujer,et al.  Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces , 2007, 0709.4363.

[4]  Alfonso Romero,et al.  Generalized maximal surfaces in Lorentz–Minkowski space L3 , 1992, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  S. Yau.  Harmonic functions on complete riemannian manifolds , 1975 .

[6]  Y. Xin.  On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space , 1991 .

[7]  Shing-Tung Yau,et al.  Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces , 1976 .

[8]  Alfonso Romero,et al.  On the Gauss curvature of maximal surfaces in the 3-dimensional Lorentz-Minkowski space , 1994 .

[9]  B. O'neill.  Semi-Riemannian Geometry With Applications to Relativity , 1983 .

[10]  L. Alías,et al.  Spacelike hypersurfaces of constant mean curvature in certain spacetimes , 1997 .

[11]  Alfonso Romero,et al.  New examples of Calabi-Bernstein problems for some nonlinear equations , 2001 .

[12]  T. Ishihara.  Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature. , 1988 .

[13]  B. Palmer.  The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space , 1990 .

[14]  L. Alías,et al.  On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space , 2000 .

[15]  Alfonso Romero,et al.  Simple proof of Calabi-Bernstein’s Theorem on maximal surfaces , 1996 .

[16]  Bennett Palmer,et al.  On the Gaussian Curvature of Maximal Surfaces and the Calabi–Bernstein Theorem , 2001 .

[17]  J. Marsden,et al.  Maximal hypersurfaces and foliations of constant mean curvature in general relativity , 1980 .

[18]  Hideki. Omori.  Isometric immersions of Riemannian manifolds , 1967 .

[19]  A. Romero,et al.  Uniqueness of Noncompact Spacelike Hypersurfaces of Constant Mean Curvature in Generalized Robertson–Walker Spacetimes , 2002 .

[20]  Miguel Sánchez,et al.  Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes , 1995 .