Bernstein-type theorems in semi-Riemannian warped products

Camargo, F.; Caminha, A.; de Lima, H.

doi:10.1090/S0002-9939-2010-10597-X

Bernstein-type theorems in semi-Riemannian warped products
HTML articles powered by AMS MathViewer

by F. Camargo, A. Caminha and H. de Lima PDF

Proc. Amer. Math. Soc. 139 (2011), 1841-1850 Request permission

Abstract:

This paper deals with complete hypersurfaces immersed in the $(n+1)$-dimensional hyperbolic and steady state spaces. By applying a technique of S. T. Yau and imposing suitable conditions on both the $r$-th mean curvatures and on the norm of the gradient of the height function, we obtain Bernstein-type results in each of these ambient spaces.

References

Alma L. Albujer and Luis J. Alías, Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Amer. Math. Soc. 137 (2009), no. 2, 711–721. MR 2448594, DOI 10.1090/S0002-9939-08-09546-4
Luis J. Alías and A. Gervasio Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 703–729. MR 2373968, DOI 10.1017/S0305004107000576
Luis J. Alías and Marcos Dajczer, Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81 (2006), no. 3, 653–663. MR 2250858, DOI 10.4171/CMH/68
Luis J. Alías, Marcos Dajczer, and Jaime Ripoll, A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Global Anal. Geom. 31 (2007), no. 4, 363–373. MR 2325221, DOI 10.1007/s10455-006-9045-5
Luis J. Alías, Alfonso Romero, and Miguel Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), no. 1, 71–84. MR 1310212, DOI 10.1007/BF02105675
João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. MR 1456513, DOI 10.1023/A:1006514303828
H. Bondi ; T. Gold. On the generation of magnetism by fluid motion. Monthly Not. Roy. Astr. Soc. 108 (1948), 252-270.
A. Caminha and H. F. de Lima, Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 1, 91–105. MR 2498961
A. Caminha, P. Sousa ; F. Camargo. Complete foliations of space forms by hypersurfaces. Accepted for publication in Bull. Braz. Math. Soc.
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
F. Hoyle. A new model for the expanding universe. Monthly Not. Roy. Astr. Soc. 108 (1948), 372-382.
Sebastián Montiel, 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 (1988), no. 4, 909–917. MR 982837, DOI 10.1512/iumj.1988.37.37045
Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711–748. MR 1722814, DOI 10.1512/iumj.1999.48.1562
Sebastián Montiel, Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes, Math. Ann. 314 (1999), no. 3, 529–553. MR 1704548, DOI 10.1007/s002080050306
Sebastián Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan 55 (2003), no. 4, 915–938. MR 2003752, DOI 10.2969/jmsj/1191418756
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211–239. MR 1216008
S. Weinberg. Gravitation and cosmology: Principles and applications of the general theory of relativity. John Wiley & Sons, New York (1972).
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051