Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The value distribution of the hyperbolic Gauss map
HTML articles powered by AMS MathViewer

by Zu-Huan Yu
Proc. Amer. Math. Soc. 125 (1997), 2997-3001
DOI: https://doi.org/10.1090/S0002-9939-97-03937-3
PDF | Request permission

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
  • R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Asterisque 154–155 (1987), 321–347.
  • Hirotaka Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geom. 29 (1989), no. 2, 245–262. MR 982173
  • Robert Osserman, Minimal surfaces in the large, Comment. Math. Helv. 35 (1961), 65–76. MR 123248, DOI 10.1007/BF02567006
  • Masaaki Umehara and Kotaro Yamada, Complete surfaces of constant mean curvature $1$ in the hyperbolic $3$-space, Ann. of Math. (2) 137 (1993), no. 3, 611–638. MR 1217349, DOI 10.2307/2946533
  • Masaaki Umehara and Kotaro Yamada, A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in $\mathbf R^3$ into the hyperbolic $3$-space, J. Reine Angew. Math. 432 (1992), 93–116. MR 1184761, DOI 10.1515/crll.1992.432.93
  • —, A duality on CMC-$1$ surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality, (preprint).
  • S. P. Wang and S. Walter Wei, Bernstein conjecture in hyperbolic geometry, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 339–358. MR 795247
  • Frederico Xavier, Erratum to the paper: “The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere” [Ann. of Math. (2) 113 (1981), no. 1, 211–214; MR 82b:53015], Ann. of Math. (2) 115 (1982), no. 3, 667. MR 657243, DOI 10.2307/2007017
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
Bibliographic 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
  • 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
  • © Copyright 1997 American Mathematical Society
  • 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