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Half-space theorems for mean curvature one surfaces in hyperbolic space

Author(s): Lucio Rodriguez; Harold Rosenberg
Journal: Proc. Amer. Math. Soc. 126 (1998), 2755-2762.
MSC (1991): Primary 53A10
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Abstract: We give conditions which oblige properly embedded constant mean curvature one surfaces in hyperbolic 3-space to intersect. Our results are inspired by the theorem that two disjoint properly immersed minimal surfaces in $\mathbf{R}^3$ must be planes.

References:

[A-R]
H. Alencar and H. Rosenberg, Some remarks on the existence of hypersurfaces of constant mean curvature with a given boundary, or asymptotic boundary, in hyperbolic space, Bull. des Sciences Maths. de France 121 (1997), 61-69.

[B]
B. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 341-347.

[C]
P. Castillon, Sur le surfaces de révolution à courbure moyenne constante dans l'espace hyperbolique, Preprint.

[doC-L]
M. do Carmo and B. Lawson, On Alexander-Bernstein theorems in hyperbolic space, Duke Math. J. 50 (1983), 995-1003. MR 85f:53009

[H-M]
D. Hoffman and W. Meeks, The strong half-space theorem for minimal surfaces, Invent. Math. 101 (1990), 373-377. MR 92e:53010

[L-R]
G. Levitt and H. Rosenberg, Symmetries of constant mean curvature hypersurfaces in hyperbolic space, Duke Math. J. 52 (1985), 53-59. MR 86h:53063

[S]
A. Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math. Ann. 277 (1987), 629-638.

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Additional Information:

Lucio Rodriguez
Affiliation: Institute for Pure-Applied Mathematics, Estrada Dona Castorina 110, 22460 Rio de Janeiro, Brazil
Email: lucio@impa.br

Harold Rosenberg
Affiliation: Department of Mathematics, University of Paris VII, 2 place Jussieu, 75251 Paris, France
Email: rosen@math.jussieu.fr

DOI: 10.1090/S0002-9939-98-04510-9
PII: S 0002-9939(98)04510-9
Received by editor(s): September 10, 1996
Communicated by: Peter Li
Copyright of article: Copyright 1998, American Mathematical Society

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