×

Constant mean curvature surfaces with two ends in hyperbolic space. (English) Zbl 0980.53081

Authors’ abstract: We investigate the close relationship between minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space. Just as in the case of minimal surfaces in Euclidean three-space, the only complete connected embedded surfaces of constant mean curvature 1 with two ends in hyperbolic space are well-understood surfaces of revolution: the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean three-space, there do exist complete connected immersed surfaces of constant mean curvature 1 with two ends in hyperbolic space that are not surfaces of revolution: the genus-one catenoid cousins. These surfaces are of interest because they show that, although minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space are intimately related, there are essential differences between these two sets of surfaces. The proof we give of existence of the genus-one catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML EMIS

References:

[1] Berglund J., Pacific J. Math. 171 (2) pp 353– (1995)
[2] Bryant R. L., Théorie des variétés minimales et applications pp 321– (1983)
[3] do Carmo M. P., Duke Math. J. 50 (4) pp 995– (1983) · Zbl 0534.53049
[4] do Carmo M. P., Comment. Math. Helv. 61 (3) pp 429– (1986)
[5] Hass J., Electron. Res. Announc. Amer. Math. Soc. 1 (3) pp 98– (1995) · Zbl 0864.53007
[6] Kapouleas N., Ann. of Math. (2) 131 (2) pp 239– (1990) · Zbl 0699.53007
[7] Karcher H., Manuscripta Math. 64 (3) pp 291– (1989) · Zbl 0687.53010
[8] Karcher H., J. Differential Geom. 28 (2) pp 169– (1988)
[9] Korevaar N. J., J. Differential Geom. 30 (2) pp 465– (1989)
[10] Korevaar N. J., Amer. J. Math. 114 (1) pp 1– (1992) · Zbl 0757.53032
[11] Levitt G., Duke Math. J. 52 (1) pp 53– (1985) · Zbl 0584.53027
[12] Osserman R., A survey of minimal surfaces (1969) · Zbl 0209.52901
[13] Polthier K., Workshop on Theoretical and Numerical Aspects of Geometric Variational Problems pp 201– (1990)
[14] Rossman W., Tohoku Math. J. 49 pp 449– (1997) · Zbl 0913.53025
[15] Schoen R. M., J. Differential Geom. 18 (4) pp 791– (1983) · Zbl 0575.53037
[16] Umehara M., J. Reine Angew. Math. 432 pp 93– (1992)
[17] Umehara M., Ann. of Math. (2) 137 (3) pp 611– (1993) · Zbl 0795.53006
[18] Umehara M., Math. Ann. 304 (2) pp 203– (1996) · Zbl 0841.53050
[19] Umehara M., Tsukuba J. Math. 21 (1) pp 229– (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.