Abstract
We derive, for the square operator of Yau, an analogue of the Omori–Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete noncompact spacelike hypersurfaces immersed with constant higher order mean curvature in a conformally stationary Lorentz manifold.
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Akutagawa K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z. 196, 13–19 (1987)
Alías L.J., Brasil A., Colares A.G. Jr.: Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications. Proc. Edinb. Math. Soc. 46, 465–488 (2003)
Alías L.J., Colares A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson–Walker spacetimes. Math. Proc. Camb. Phil. Soc. 143, 703–729 (2007)
Alías L.J., Romero A., Sánchez M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 27, 71–84 (1995)
Barbosa J.L.M., Colares A.G.: Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15, 277–297 (1997)
Brasil A., Colares A.G. Jr., Palmas O.: Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: a gap theorem. Illinois J. Math. 47(3), 847–866 (2003)
Camargo F.E.C., Chaves R.B., de Sousa L.A.M., Jr. (2008) Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space. Diff. Geom. Appl. (to appear)
Caminha A.: On spacelike hypersurfaces of constant sectional curvature lorentz manifolds. J. Geom. Phys. 56, 1144–1174 (2006)
Caminha A.: A rigidity theorem for complete CMC hypersurfaces in lorentz manifolds. Diff. Geom. Appl. 24, 652–659 (2006)
Cheng S.Y., Yau S.T.: Maximal spacelike hypersurfaces in the Lorentz–Minkowski space. Ann. Math. 104, 407–419 (1976)
Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
do Carmo M.: Riemannian geometry. Birkhauser, Boston (1992)
Goddard A.J.: Some remarks on the existence of spacelike hypersurfaces with constant mean curvature. Math. Proc. Camb. Phil. Soc. 82, 489–495 (1977)
Hardy G., Littlewood J.E., Pólya G.: Inequalities. Cambridge Mathematical Library, Cambridge (1989)
Hawking S.W., Ellis G.F.R.: The large scale structure of spacetime. Cambridge University Press, Cambridge (1973)
Hounie J., Leite M.L.: Two-ended hypersurfaces with zero scalar curvature. Indiana Univ. Math. J. 48, 867–882 (1999)
de Lima H.F.: Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space. J. Geom. Phys. 57, 967–975 (2007)
Marsden J., Tipler F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Bull. Am. Phys. Soc. 23, 84 (1978)
Montiel S.: An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature. Indiana Univ. Math. J. 37, 909–917 (1988)
Montiel S.: Uniqueness of Spacelike Hypersurfaces of Constant Mean Curvature in foliated Spacetimes. Math. Ann. 314, 529–553 (1999)
Montiel S.: Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces. J. Math. Soc. Jpn. 55, 915–938 (2003)
O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)
Schoen R, Yau S.T.: Lectures on Differential Geometry. International Press Inc., Cambridge (1994)
Stumbles S.: Hypersurfaces of constant mean extrinsic curvature. Ann. Phys. 133, 28–56 (1980)
Wald R.: General Relativity. Univ of Chicago Press, Chicago (1984)
Yau S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)
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Caminha, A., de Lima, H.F. Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds. Gen Relativ Gravit 41, 173–189 (2009). https://doi.org/10.1007/s10714-008-0663-z
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DOI: https://doi.org/10.1007/s10714-008-0663-z