Abstract
We apply suitable generalized maximum principles in order to obtain new Calabi–Bernstein’s type results concerning complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime. Assuming a natural comparison inequality between the weighted mean curvatures of the hypersurface and those of the slices of the timelike bounded region where the hypersurface is supposed to be contained, we get sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, we also treat the case when the ambient spacetime is static.
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Albujer A.L., Camargo F., de Lima H.F.: Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Cambridge Philos. Soc., 151, 271–282 (2011)
Alías L.J., Colares A.G.: Uniqueness of spacelike hypersurface with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Cambridge Philos. Soc., 143, 703–729 (2007)
L. J. Alías, D. Impera and M. Rigoli, Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes, Math. Proc. Cambridge Philos. Soc., 152 (2012), 365–383.
L. J. Alías, A. Romero and M. Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, Gen. Relat. Grav., 27 (1995), 71–84.
D. Bakry and M. Émery, Diffusions hypercontractives, in: Seminaire de probabilites, XIX, 1983/84, volume 1123 of Lecture Notes in Mathematics, Springer (Berlin, 1985), pp. 177–206.
M. Caballero, A. Romero and R. M. Rubio, Uniqueness of maximal surfaces in generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems, J. Geom. Phys., 60 (2010), 394–402.
M. Caballero, A. Romero and R. M. Rubio, Complete cmc spacelike surfaces with bounded hyperbolic angle in generalized Robertson–Walker spacetimes, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 961–978.
E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Sympos. Pure Math., 15 (1970), 223–230.
F. Camargo, A. Caminha, H. F. de Lima and U. Parente, Generalized maximum principles and the rigidity of complete spacelike hypersurfaces, Math. Proc. Cambridge Philos. Soc., 153 (2012), 541–556.
F. Camargo, A. Caminha, H. F. de Lima and M. Velásquez, On the geometry of conformally stationary Lorentz spaces, Acta Math. Hungar., 134 (2012), 385–403.
A. Caminha, The geometry of closed conformal vector fields on Riemannian spaces, Bull. Brazilian Math. Soc., 42 (2011), 277–300.
J. S. Case, Singularity theorems and the Lorentzian splitting theorem for the Bakry– Émery Ricci tensor, J. Geom. Phys., 60 (2010), 477–490.
S. Y. Cheng and S. T. Yau, Maximal spacelike hypersurfaces in the Lorentz–Minkowski space, Ann. of Math., 104 (1976), 407–419.
H. F. de Lima and U. L. Parente, On the geometry of maximal spacelike hypersurfaces in generalized Robertson–Walker spacetimes, Ann. Mat. P. Appl., 192 (2013), 649–643.
M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178–215.
G. Huang, C. Zhang and J. Zhang, Liouville-type theorem for the drifting Laplacian operator, Arch. Math., 96 (2011), 379–385.
H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19 (1967), 205–214.
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press (London, 1983).
M. Rimoldi, Rigidity results for Lichnerowicz Bakry–Emery Ricci tensor, Ph.D. thesis, Università degli Studi di Milano (Milano, 2011).
A. Romero and R. M. Rubio, On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems, Ann. Glob. Anal. Geom., 37 (2010), 21–31.
A. Romero, R. M. Rubio and J. J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes, Class. Quantum Grav., 30 (2013), 115007 (13pp).
A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39–56.
G. Wei and W. Wylie, Comparison geometry for the Bakry–Émery Ricci tensor, J. Diff. Geom., 83 (2009), 377–405.
S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201–228.
S.-T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659–670.
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The first author is partially supported by CNPq/Brazil, grant 306131/2012-9. The second is partially supported by CNPq, Brazil, grant 300769/2012-1. The third author is partially supported by CAPES/Brazil.
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Cavalcante, M.P., de Lima, H.F. & Santos, M.S. New Calabi–Bernstein type results in weighted generalized Robertson–Walker spacetimes. Acta Math. Hungar. 145, 440–454 (2015). https://doi.org/10.1007/s10474-014-0461-x
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DOI: https://doi.org/10.1007/s10474-014-0461-x
Key words and phrases
- generalized Robertson–Walker spacetime
- weighted manifold
- Bakry–Émery Ricci tensor
- drifting Laplacian
- weighted mean curvature
- complete spacelike hypersurface