Proceedings of the Japan Academy, Series A, Mathematical Sciences

CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere

Shoichi Fujimori, Yu Kawakami, Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, and Kotaro Yamada

Full-text: Open access

Abstract

CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero with three regular embedded ends) in hyperbolic 3-space $H^{3}$ are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so here we give an explicit description of CMC-1 trinoids in $H^{3}$ that includes the reducible case.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 8 (2011), 144-149.

Dates
First available in Project Euclid: 3 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1317647397

Digital Object Identifier
doi:10.3792/pjaa.87.144

Mathematical Reviews number (MathSciNet)
MR2843096

Zentralblatt MATH identifier
1242.53070

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53A35: Non-Euclidean differential geometry
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 33C05: Classical hypergeometric functions, $_2F_1$

Keywords
Constant mean curvature spherical metrics conical singularities trinoids

Citation

Fujimori, Shoichi; Kawakami, Yu; Kokubu, Masatoshi; Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro. CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 8, 144--149. doi:10.3792/pjaa.87.144. https://projecteuclid.org/euclid.pja/1317647397


Export citation

References

  • A. I. Bobenko, T. V. Pavlyukevich and B. A. Springborn, Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions, Math. Z. 245 (2003), no. 1, 63–91.
  • B. Daniel, Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space, J. Differential Geom. 72 (2006), no. 3, 467–508.
  • A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3349–3355.
  • M. Furuta and Y. Hattori, 2-dimensional singular spherical space forms, manuscript, 1998.
  • W. Rossman, M. Umehara and K. Yamada, Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. I, Hiroshima Math. J. 34 (2004), no. 1, 21–56.
  • W. Rossman, M. Umehara and K. Yamada, Period problems for mean curvature one surfaces in $H^{3}$, Surveys on Geometry and Integrable systems, Advanced Studies in Pure Mathematics 51 (2008), 347–399.
  • M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821.
  • M. Umehara and K. Yamada, A duality on CMC-1 surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality, Tsukuba J. Math. 21 (1997), no. 1, 229–237.
  • M. Umehara and K. Yamada, Metrics of constant curvature 1 with three conical singularities on the 2-sphere, Illinois J. Math. 44 (2000), no. 1, 72–94.