Tohoku Mathematical Journal

Kenmotsu type representation formula for surfaces with prescribed mean curvature in the $3$-sphere

Reiko Aiyama and Kazuo Akutagawa

Full-text: Open access

Abstract

Our primary object of this paper is to give a representation formula for a surface with prescribed mean curvature in the (metric) 3-sphere by means of a single component of the generalized Gauss map. For a CMC (constant mean curvature) surface, we derive another representation formula by means of the adjusted Gauss map. These formulas are spherical versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. Spin versions of them are obtained as well.

Article information

Source
Tohoku Math. J. (2), Volume 52, Number 1 (2000), 95-105.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178224660

Digital Object Identifier
doi:10.2748/tmj/1178224660

Mathematical Reviews number (MathSciNet)
MR1740545

Zentralblatt MATH identifier
1008.53012

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53C27: Spin and Spin$^c$ geometry

Citation

Aiyama, Reiko; Akutagawa, Kazuo. Kenmotsu type representation formula for surfaces with prescribed mean curvature in the $3$-sphere. Tohoku Math. J. (2) 52 (2000), no. 1, 95--105. doi:10.2748/tmj/1178224660. https://projecteuclid.org/euclid.tmj/1178224660


Export citation

References

  • [1] R AIYAMA AND K AKUTAGAWA, Kenmotsu-Bryant type representation formula for constant mean curva-ture surfaces in H3(-c2) and S^(c2), Ann Global Anal Geom. 17 (1999), 49-75.
  • [2] R AIYAMA AND K AKUTAGAWA, Kenmotsu type representation formula for surfaces with prescribed mea curvature in the hyperbolic 3-space, to appear in J Math Soc Japan.
  • [3] R AIYAMA, K AKUTAGAWA, R. MIYAOKA AND M. UMEHARA, A global correspondence between CMC surfaces in S3 and pairs of non-holomorphic harmonic maps into S2, Proc. Amer Math Soc. 128 (2000), 939-941
  • [4] A I. BOBENKO, Constant mean curvature surfaces andintegrable equations, Russian Math Survey 46 (1991), 1-45
  • [5] R. L BRYANT, Surfaces of mean curvature one in hyperbolic space, Asterisque 154-155 (1987), 321-347
  • [6] A FUJIOKA, Harmonic maps and associated maps from simply connected Riemann surfaces into the 3 dimensional space forms, Thoku Math. J. 47 (1995), 431-439.
  • [7] D A HOFFMAN AND R OSSERMAN, The Gauss mapof surfaces in R3 and/?4, Proc. London Math Soc 50 (1985), 27-56
  • [8] K KENMOTSU, Weierstrass formula for surfaces of prescribed mean curvature, Math Ann 245 (1979), 89 99
  • [9] B KONOPELCHENKO AND I TAIMANOV, Constant mean curvature surfaces via an integrable dynamica system, ! Phys A 29 (1996), 1261-1265.
  • [10] R KUSNER AND N SCHMITT, The spinor representation of surfaces in space, E-print dg-ga/961000
  • [11] B LAWSON, Complete minimal surfaces in S3, Ann. of Math. 92 (1970), 335-37
  • [12] R MIYAOKA, The splitting and deformations of the generalized Gauss map of compact CMC surfaces, Tohoku Math. J. 51 (1999), 35-53.
  • [13] J MORGAN, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mat Notes 44, Princeton Univ Press, 1996
  • [14] W SEAMAN, On surfaces in R4, Proc. Amer. Math Soc 94 (1985), 467-47
  • [15] M UMEHARA AND K YAMADA, A parametrization of the Weierstrass formulae and perturbation of complet minimal surfaces in/? into the hyperbolic 3-spaces, J Reine Angew Math 432 (1992), 93-116