New constant mean curvature surfaces. (English) Zbl 1017.53009

Authors’ abstract: We use the Dorfmeister-Pedit-Wu construction to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the third named author, describes the DPW potentials that appear to give CMC punctured spheres with \(k\) Delaunay ends \((k\)-noids): the evidence is experimental at present. These can have both unduloidal an nodoidal ends.
Reviewer: D.Ferus (Berlin)


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI Euclid EuDML


[1] Bobenko A. I., Uspekhi Mat. Nauk 46 (4) pp 3– (1991)
[2] Bobenko A. I., Harmonic maps and integrable systems 23 pp 83– (1994) · Zbl 0841.53003
[3] Burstall F. E., Harmonic maps and integrable systems pp 221– (1994)
[4] DOI: 10.1215/S0012-7094-95-08015-6 · Zbl 0966.58007
[5] Dorfmeister J., ”Construction of non-simply connected CMC surfaces via dressing” · Zbl 1035.53015
[6] Dorfmeister J., J. Reine Angew. Math. 440 pp 43– (1993)
[7] Dorfmeister J., Comm. Anal. Geom. 6 (4) pp 633– (1998)
[8] DOI: 10.1007/BF02572424 · Zbl 0806.53005
[9] Gro K., ”Constant mean curvature surfaces with three ends” (1999)
[10] Korevaar N. J., J. Differential Geom. 30 (2) pp 465– (1989)
[11] DOI: 10.1007/978-1-4613-9711-3_11
[12] Lerner, D. and Sterling, I. ”Construction of constant mean curvature surfaces using the Dorfmeister–Pedit–Wu representation of harmonic maps”. Proceedings of the Symposium on Differential Geometry, Hamiltonian Systems, and Operator Theory. 1994, Mona, Jamaica. The University Printers (Univ. of the West Indies). [Lerner and Sterling 1995]
[13] Press W. H., Numerical recipes in C,, 2. ed. (1992) · Zbl 0778.65003
[14] Pressley A., Loop groups (1986) · Zbl 0618.22011
[15] DOI: 10.1007/978-1-4613-8324-6_9
[16] DOI: 10.1512/iumj.1993.42.42057 · Zbl 0803.53009
[17] DOI: 10.1007/BF02571730 · Zbl 0799.53011
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.