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On the rigidity of constant mean curvature complete vertical graphs in warped products. (English) Zbl 1219.53056
Summary: We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriate convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning such graphs. Furthermore, applications to hyperbolic and Euclidean spaces are given.

##### MSC:
 53C42 Immersions (differential geometry) 53C24 Rigidity results (differential geometry)
##### References:
 [1] Akutagawa, K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196, 13-19 (1987) · Zbl 0611.53047 · doi:10.1007/BF01179263 [2] Alías, L. J.; Dajczer, M.: Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. math. Helv. 81, 653-663 (2006) · Zbl 1110.53039 · doi:10.4171/CMH/68 [3] Alías, L. J.; Dajczer, M.; Ripoll, J.: A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. global anal. Geom. 31, 363-373 (2007) · Zbl 1125.53005 · doi:10.1007/s10455-006-9045-5 [4] Alías, L. J.; Romero, A.; Sánchez, M.: Uniqueness of complete spacelike hypersurfaces with constant mean curvature in generalized Robertson-Walker spacetimes, Gen. relativity gravitation 27, 71-84 (1995) · Zbl 0908.53034 · doi:10.1007/BF02105675 [5] C.P. Aquino, H.F. de Lima, On the Gauss map of complete CMC hypersurfaces in the hyperbolic space, J. Math. Anal. Appl., in press. [6] Camargo, F.; Caminha, A.; De Lima, H. F.: Bernstein-type theorems in semi-Riemannian warped products, Proc. amer. Math. soc. 139, 1841-1850 (2011) · Zbl 1223.53045 · doi:10.1090/S0002-9939-2010-10597-X [7] A. Caminha, The geometry of closed conformal vector fields on Riemannian spaces, Bull. Brazilian Math. Soc., in press. [8] Caminha, A.; De Lima, H. F.: Complete vertical graph with constant mean curvature in semi-Riemannian warped products, Bull. belg. Math. soc. 16, 91-105 (2009) · Zbl 1160.53362 · doi:euclid:bbms/1235574194 [9] Montiel, S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana univ. Math. J. 48, 711-748 (1999) · Zbl 0973.53048 · doi:10.1512/iumj.1999.48.1562 · doi:http://www.iumj.indiana.edu/TOC/992.htm [10] Omori, H.: Isometric immersions of Riemannian manifolds, J. math. Soc. Japan 19, 205-214 (1967) · Zbl 0154.21501 · doi:10.2969/jmsj/01920205 [11] O’neill, B.: Semi-Riemannian geometry, with applications to relativity, (1983) [12] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Comm. pure appl. Math. 28, 201-228 (1975) · Zbl 0291.31002 · doi:10.1002/cpa.3160280203 [13] Yau, S. T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana univ. Math. J. 25, 659-670 (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051