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

Authors:
Lucio Rodriguez and Harold Rosenberg

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2755-2762

MSC (1991):
Primary 53A10

DOI:
https://doi.org/10.1090/S0002-9939-98-04510-9

MathSciNet review:
1458259

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 must be planes.

**[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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
53A10

Retrieve articles in all journals with MSC (1991): 53A10

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:
https://doi.org/10.1090/S0002-9939-98-04510-9

Received by editor(s):
September 10, 1996

Communicated by:
Peter Li

Article copyright:
© Copyright 1998
American Mathematical Society