Tokyo Journal of Mathematics

Surfaces and Fronts with Harmonic-mean Curvature One in Hyperbolic Three-space

Masatoshi KOKUBU

Abstract

We show a global representation formula for a certain kind of Weingarten surface in hyperbolic three-space, which is based on the formula due to Gálvez, Martínez and Milán. As an application of the representation formula, we also investigate surfaces with harmonic-mean curvature one (HMC-1 surfaces). We allow them to have certain kinds of singularities, and discuss some global properties.

Article information

Source
Tokyo J. Math., Volume 32, Number 1 (2009), 177-200.

Dates
First available in Project Euclid: 7 August 2009

https://projecteuclid.org/euclid.tjm/1249648416

Digital Object Identifier
doi:10.3836/tjm/1249648416

Mathematical Reviews number (MathSciNet)
MR2541163

Zentralblatt MATH identifier
1211.53079

Citation

KOKUBU, Masatoshi. Surfaces and Fronts with Harmonic-mean Curvature One in Hyperbolic Three-space. Tokyo J. Math. 32 (2009), no. 1, 177--200. doi:10.3836/tjm/1249648416. https://projecteuclid.org/euclid.tjm/1249648416

References

• R. Bryant, Surfaces of mean curvature one in hyperbolic space, in Théorie des variétés minimales et applications, Astérisque, 154–155 (1988), 321–347.
• P. Collin, L. Hauswirth and H. Rosenberg, The geometry of finite topology Bryant surfaces, Ann. of Math. (2), 153 (2001), no. 3, 623–659.
• C. L. Epstein, Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic $3$-Space, unpublished.,
• J. A. Gálvez, A. Martínez and F. Milán, Flat surfaces in hyperbolic $3$-space, Math. Ann., 316 (2000), no. 3, 419–435.
• J. A. Gálvez, A. Martínez and F. Milán, Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity, Trans. Amer. Math. Soc., 356 (2004), no. 9, 3405–3428.
• M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math., 221 (2005), no. 2, 303–351.
• M. Kokubu, M. Umehara and K. Yamada, An elementary proof of Small's formula for null curves in $\PSL(2,\C)$ and an analogue for Legendrian curves in $\PSL(2,\C)$, Osaka J. Math., 40 (2003), no. 3, 697–715.
• M. Kokubu, M. Umehara and K. Yamada, Flat fronts in hyperbolic $3$-space, Pacific J. Math., 216 (2004), no. 1, 149–175.
• M. Spivak, A comprehensive introduction to differential geometry, Vol. V, Publish or Perish, Inc., 1979.
• M. Umehara and K. Yamada, Complete surfaces of constant mean curvature $1$ in the hyperbolic $3$-space, Ann. of Math. (2), 137 (1993), no. 3, 611–638.