Coarse classification of constant mean curvature cylinders
HTML articles powered by AMS MathViewer
- by J. Dorfmeister and S.-P. Kobayashi PDF
- Trans. Amer. Math. Soc. 359 (2007), 2483-2500 Request permission
Abstract:
We give a coarse classification of constant mean curvature (CMC) immersions of cylinders into $\mathbb {R}^3$ via the loop group method. Particularly for this purpose, we consider double loop groups and a new type of “potentials” which are meromorphic 1-forms on Riemann surfaces.References
- A. I. Bobenko, Surfaces of constant mean curvature and integrable equations, Uspekhi Mat. Nauk 46 (1991), no. 4(280), 3–42, 192 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 4, 1–45. MR 1138951, DOI 10.1070/RM1991v046n04ABEH002826
- M. J. Bergvelt and M. A. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc. 326 (1991), no. 2, 861–886. MR 1062870, DOI 10.1090/S0002-9947-1991-1062870-5
- Frank Bowman, Introduction to Bessel functions, Dover Publications, Inc., New York, 1958. MR 0097539
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- J. Dorfmeister, Generalized Weierstrass representations of surfaces, Surveys on Geometry and Integrable Systems, Advanced Studies in Pure Mathematics, to appear.
- J. Dorfmeister and G. Haak, Meromorphic potentials and smooth surfaces of constant mean curvature, Math. Z. 224 (1997), no. 4, 603–640. MR 1452051, DOI 10.1007/PL00004295
- J. Dorfmeister and G. Haak, On symmetries of constant mean curvature surfaces. II. Symmetries in a Weierstraß-type representation, Int. J. Math. Game Theory Algebra 10 (2000), no. 2, 121–146. MR 1784970
- J. Dorfmeister and G. Haak, On constant mean curvature surfaces with periodic metric, Pacific J. Math. 182 (1998), no. 2, 229–287. MR 1609603, DOI 10.2140/pjm.1998.182.229
- Josef Dorfmeister and Guido Haak, Construction of non-simply connected CMC surfaces via dressing, J. Math. Soc. Japan 55 (2003), no. 2, 335–364. MR 1961290, DOI 10.2969/jmsj/1191419120
- J. Dorfmeister, M. Kilian, Dressing preserving the fundamental group, preprint.
- J. Dorfmeister, S.-P. Kobayashi, F. Pedit, Complex surfaces of constant mean curvature fibered by minimal surfaces, in preparation.
- Josef Dorfmeister, Ian McIntosh, Franz Pedit, and Hongyou Wu, On the meromorphic potential for a harmonic surface in a $k$-symmetric space, Manuscripta Math. 92 (1997), no. 2, 143–152. MR 1428645, DOI 10.1007/BF02678186
- J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668. MR 1664887, DOI 10.4310/CAG.1998.v6.n4.a1
- Josef Dorfmeister and Hongyou Wu, Constant mean curvature surfaces and loop groups, J. Reine Angew. Math. 440 (1993), 43–76. MR 1225957
- Martin A. Guest, Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts, vol. 38, Cambridge University Press, Cambridge, 1997. MR 1630443, DOI 10.1017/CBO9781139174848
- R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
- Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043, DOI 10.1007/978-3-662-59158-1
- Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. An introduction to the fundamental theory; With the assistance of Gottfried Barthel; Translated from the German by Michael Bridgland. MR 716497, DOI 10.1515/9783110838350
- M. Kilian, Constant mean curvature cylinders, Doctoral Thesis, Univ. of Massachusetts (Amherst), U.S.A., September 2000.
- Martin Kilian, Ian McIntosh, and Nicholas Schmitt, New constant mean curvature surfaces, Experiment. Math. 9 (2000), no. 4, 595–611. MR 1806295, DOI 10.1080/10586458.2000.10504663
- Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
- N. Schmitt, Constant mean curvature trinoids, preprint.
- N. Schmitt, CMCLab, http://www.gang.umass.edu/software.
- Hongyou Wu, A simple way for determining the normalized potentials for harmonic maps, Ann. Global Anal. Geom. 17 (1999), no. 2, 189–199. MR 1675409, DOI 10.1023/A:1006556302766
Additional Information
- J. Dorfmeister
- Affiliation: Zentrum Mathematik, Technische Universität München Boltzmannstr. 3, D-85747, Garching, Germany
- Email: dorfm@ma.tum.de
- S.-P. Kobayashi
- Affiliation: School of Information Environment, Tokyo Denki University Muzai Gakuendai 2-1200 Inzai, Chiba 270-1382, Japan
- Email: shimpei@sie.dendai.ac.jp
- Received by editor(s): December 7, 2004
- Published electronically: January 4, 2007
- Additional Notes: The first author acknowledges support by DFG
The second author was fully supported by DFG grant DO776/1. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2483-2500
- MSC (2000): Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9947-07-04063-9
- MathSciNet review: 2286041