Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The value distribution
of the hyperbolic Gauss map


Author: Zu-Huan Yu
Journal: Proc. Amer. Math. Soc. 125 (1997), 2997-3001
MSC (1991): Primary 53A10; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-97-03937-3
MathSciNet review: 1401758
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in $H^3(-1)$, and prove that it cannot omit more than four points unless the surface is a horosphere.


References [Enhancements On Off] (What's this?)

  • 1. R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Asterisque 154-155 (1987), 321-347. CMP 20:16
  • 2. H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Diff. Geo. 29 (1989), 245-262. MR 89m:53012
  • 3. R. Osserman, Minimal surfaces in the large, Comment. Math. Helv. 35 (1961), 65-76. MR 23:A576
  • 4. M. Umehara and K. Yamada, Complete surfaces of constant mean curvature-$1$ in the hyperbolic $3$-space, Ann. of Math. 137 (1993), 611-638. MR 94c:53015
  • 5. M. Umehara and K. Yamada, A parametrization of the Weierstrass formulae minimal surfaces in $R^3$ into the hyperbolic $3$-space, J. Reine Angew. Math. 432 (1992), 93-116. MR 94e:54004
  • 6. -, A duality on CMC-$1$ surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality, (preprint).
  • 7. S. P. Wang and S. Walter Wei, Bernstein conjecture in hyperbolic geometry, Seminar on minimal submanifolds, Ann. Math. Studies 103, Princeton Univ. Press, 1983. MR 86m:53076
  • 8. F. Xavier, The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere, Ann. of Math. (2) 133 (1981), 211-214. MR 83h:53016

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53A10, 53C42

Retrieve articles in all journals with MSC (1991): 53A10, 53C42


Additional Information

Zu-Huan Yu
Affiliation: Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
Address at time of publication: Fundamental Department, Jiaozuo Institute of Technology, Jiaozuo 454159, Henan Province, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-97-03937-3
Keywords: Constant mean curvature, hyperbolic space, Gauss map.
Received by editor(s): November 1, 1995
Received by editor(s) in revised form: April 2, 1996
Additional Notes: Partially supported by NNSFC and SFECC
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society