Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Complete minimal surfaces in $ {\bf R}\sp 3$ of genus one and four planar embedded ends


Author: Celso J. Costa
Journal: Proc. Amer. Math. Soc. 119 (1993), 1279-1287
MSC: Primary 53A10; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1993-1160295-2
MathSciNet review: 1160295
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By using elliptic functions and Weierstrass representation we construct a one-parameter family of complete minimal surfaces in $ {{\mathbf{R}}^3}$ with genus one and four planar embedded ends. These surfaces are critical points of the Willmore functional.


References [Enhancements On Off] (What's this?)

  • [1] R. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 23-53. MR 772125 (86j:58029)
  • [2] C. J. Costa, Example of a complete minimal immersion of genus one and three embedded ends, Boll. Soc. Brasil. Mat. 15 (1984), 47-54. MR 794728 (87c:53111)
  • [3] -, Uniqueness of minimal surfaces embedded in $ {{\mathbf{R}}^3}$, with total curvature $ 12\pi $, J. Differential Geom. 30 (1989), 597-618. MR 1021368 (90k:53011)
  • [4] -, Classification of complete minimal surfaces in $ {{\mathbf{R}}^3}$ with total curvature $ 12\pi $, Invent. Math. 105 (1991), 273-303. MR 1115544 (92h:53010)
  • [5] D. Hoffman and W. Meeks, A complete embedded minimal surface in $ {{\mathbf{R}}^3}$ with genus one and three ends, J. Differential Geom. 21 (1985), 109-127. MR 806705 (87d:53008)
  • [6] -, The strong half-space theorem for minimal surfaces, Invent. Math. 101 (1990), 373-377. MR 1062966 (92e:53010)
  • [7] L. P. Jorge and W. Meeks, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203-221. MR 683761 (84d:53006)
  • [8] R. Kusner, Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (1989), 317-345. MR 996204 (90e:53013)
  • [9] -, Doctoral Thesis, Univ. Calif., Berkeley, 1985.
  • [10] S. Montiel and A. Ros, Index of complete minimal surfaces, Schrödinger operator associated with a holomorphic map, preprint.
  • [11] R. Osserman, A survey of minimal surfaces, Van Nostrand Reinhold, New York, 1969. MR 0256278 (41:934)
  • [12] C. Peng, Some new examples of minimal surfaces in $ {{\mathbf{R}}^3}$ and its application, MSRI 07510-85, preprint.
  • [13] H. Rosenberg and E. Toubiana, Some remarks on deformation of minimal surfaces, Trans. Amer. Math. Soc. 295 (1986), 476-489. MR 833693 (88a:53005b)
  • [14] R. Schoen, Uniqueness, symmetry and embeddeness of minimal surfaces, J. Differential Geom. 18 (1983), 791-809. MR 730928 (85f:53011)
  • [15] J. Tannery and R. Molk, Éléments de la théorie des fonctions elliptiques, Chelsea, New York, 1972.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A10, 53C42

Retrieve articles in all journals with MSC: 53A10, 53C42


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1160295-2
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society