Tohoku Mathematical Journal

Irreducible constant mean curvature $1$ surfaces in hyperbolic space with positive genus

Wayne Rossman, Masaaki Umehara, and Kotaro Yamada

Full-text: Open access

Article information

Source
Tohoku Math. J. (2), Volume 49, Number 4 (1997), 449-484.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178225055

Digital Object Identifier
doi:10.2748/tmj/1178225055

Mathematical Reviews number (MathSciNet)
MR1478909

Zentralblatt MATH identifier
0913.53025

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53A35: Non-Euclidean differential geometry 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro. Irreducible constant mean curvature $1$ surfaces in hyperbolic space with positive genus. Tohoku Math. J. (2) 49 (1997), no. 4, 449--484. doi:10.2748/tmj/1178225055. https://projecteuclid.org/euclid.tmj/1178225055


Export citation

References

  • [Bou] N. BOURBAKI, GROUPES ETALGEBRES DE LIE, Elements de Math., Actualites Sci. Indust., Hermann, Paris, 1968.
  • [Bry] R. BRYANT, Surfaces of mean curvature one in hyperbolic space, Asterisque 154-155 (1987), 341-347.
  • [BR] J. BERGLUND AND W. ROSSMAN, Minimal surfaces with catenoid ends, Pacific J. Math. 171 (1995), 353-371.
  • [JM] J. P. JORGE AND W. H. MEEKS, III, The topology of complete minimal surfaces of finite total Gaussia curvature, Topology 22 (1983), 203-221.
  • [Kar] H. KARCHER, Construction of minimal suraces, Surveys in Geometry 1989/90, University o Tokyo.
  • [Kar2] H. KARCHER, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math. 6 (1988), 83-114.
  • [Kat] S. KATO, Construction of «-end catenoids with prescribed flux, Kodai Math. J. 18 (1995), 86-98.
  • [L] H. B. LAWSON, Complete minimal surfaces in S3, Ann. of Math. 92, no. 2 (1970), 335-374
  • [LR] F. J. LOPEZ AND A. Ros, On embedded complete minimal surfaces of genus zero, J. Differentia Geometry 33 (1991), 293-300.
  • [O] R. OSSERMAN, A Survey of Minimal Surfaces, 2nd Ed., Dover Publ., 1986
  • [R] W. ROSSMAN, Minimal surfaces in R3 with dihedral symmetry. Thoku Math. J. 47 (1995), 31-54.
  • [S] A. J. SMALL, Surface of constant mean curvature 1 in 7/3 and algebraic curves on a quadric, Proc Amer. Math. Soc. 122 (1994), 1211-1220.
  • [Sa] K. SATO, Construction of higher genus minimal surfaces with one end and finite total curvature, Thoku Math. J. 48 (1996), 229-246.
  • [T] M. TROYANOV, Metrics of constant curvature on a sphere with two conical singularities, Lectur Notes in Math. 1410, Springer-Verlag, 1988, 236-308.
  • [UY1] M. UMEHARA AND K. YAMADA, Complete surfaces of constant mean curvature-1 in the hyperboli 3-space, Ann. of Math. 137 (1993), 611-638.
  • [UY2] M. UMEHARA AND K. YAMADA, A parametrization of Weierstrass formulae and perturbation o some complete minimal surfaces of R3 into the hyperbolic 3-space, J. Reine Angew. Math. 432 (1992), 93-116.
  • [UY3] M. UMEHARA AND K. YAMADA, Deformations of Lie Groups and their application to surface theory, in "Geometry and Its Applications" (T. Nagano et al., eds.), World Scientific, Singapore, 1993, 241-255.
  • [UY4] M. UMEHARA AND K. YAMADA, Surfaces of constant mean curvature-c in H3(-c2) with prescribe hyperbolic Gauss map, Math. Ann. 304 (1996), 203-224.
  • [UY5] M. UMEHARA AND K. YAMADA, A duality on CMC-1 surfaces in the hyperbolic space and hyperbolic analogue of the Osserman inequality, Tsukuba J. Math. 21 (1997), 229-237.
  • [X] Y. Xu, Symmetric minimal surfaces in /?3, Pacific J. Math. 171 (1995), 275-296
  • [W] M. WOHLGEMUTH, Higher genus minimal surfaces of finite total curvature, preprint (1994)