Bulletin of the American Mathematical Society

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ISSN 1088-9485(e) ISSN 0273-0979(p)

Adding handles to the helicoid

Author(s): David Hoffman; Fu Sheng Wei; Hermann Karcher
Journal: Bull. Amer. Math. Soc. 29 (1993), 77-84.
MSC (2000): Primary 53A10; Secondary 58E12
MathSciNet review: 1193537
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Abstract | References | Similar articles | Additional information

Abstract: There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one.

References:

[1]: M. Callahan, D. Hoffman, and J. Hoffman, Computer graphics tools for the study of minimal surfaces, Comm. ACM 31 (1988), 648-661. MR 945033 (89k:58063)
[2]: C. Costa, Example of a complete minimal immersion in ${\mathbb{R}^3}$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), 47-54. MR 794728 (87c:53111)
[3]: D. Hoffman, The discovery of new embedded minimal surfaces: elliptic functions, symmetry; computer graphics, Proceedings of the Berlin Conference on Global Differential Geometry, Lecture Notes in Math., vol. 1748, Springer-Verlag, New York, 1985. MR 824068
[4]: D. Hoffman and W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), 1-34. MR 1038356 (91i:53010)
[5]: -, Minimal surfaces based on the catenoid, Amer. Math. Monthly, Special Geometry Issue 97 (1990), 702-730. MR 1072813 (92c:53002)
[6]: D. Hoffman and M. Wohlgemuth, Limiting behavior of classical periodic minimal surfaces, GANG preprint series III (to appear).
[7]: J. T. Hoffman, MESH manual, GANG preprint series II, #35.
[8]: H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math. 62 (1988), 83-114. MR 958255 (89i:53009)
[9]: F. J. Lopez and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293-300. MR 1085145 (91k:53019)
[10]: W. H. Meeks III and H. Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989), 351-379. MR 1001845 (90m:53017)
[11]: J. C. C. Nitsche, Lectures on minimal surfaces, vol.1, Cambridge Univ. Press, Cambridge, 1989. MR 1015936 (90m:49031)
[12]: R. Osserman, Global properties of minimal surfaces in ${E^3}$ and ${E^n}$ , Ann. of Math. (2) 80 (1964), 340-364. MR 0179701 (31:3946)
[13]: -, A survey of minimal surfaces, 2nd ed., Dover, New York, 1986. MR 852409 (87j:53012)
[14]: I. Peterson, Three bites in a doughnut, Sci. News 127 (1985), 168-169.
[15]: H. F. Scherk, Bemerkungen über die kleinste fläche innerhalb gegebener grenzen, J. Reine Angew. Math. 13 (1835), 185-208.
[16]: R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), 791-809. MR 730928 (85f:53011)
[17]: F. Wei, Some existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math. 109 (1992), 113-136. MR 1168368 (93c:53004)
[18]: M. Wohlgemuth, Higher genus minimal surfaces by growing handles out of a catenoid, Manuscripta Math. 70 (1991), 397-428. MR 1092145 (91k:53021)

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