Adding handles to the helicoid
HTML articles powered by AMS MathViewer
- by David Hoffman, Fu Sheng Wei and Hermann Karcher PDF
- Bull. Amer. Math. Soc. 29 (1993), 77-84 Request permission
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
- Michael J. Callahan, David Hoffman, and James T. Hoffman, Computer graphics tools for the study of minimal surfaces, Comm. ACM 31 (1988), no. 6, 648–661. MR 945033, DOI 10.1145/62959.62961
- Celso J. Costa, Example of a complete minimal immersion in $\textbf {R}^3$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), no. 1-2, 47–54. MR 794728, DOI 10.1007/BF02584707
- David Hoffman, Embedded minimal surfaces, computer graphics and elliptic functions, Global differential geometry and global analysis 1984 (Berlin, 1984) Lecture Notes in Math., vol. 1156, Springer, Berlin, 1985, pp. 204–215. MR 824068, DOI 10.1007/BFb0075092
- David Hoffman and William H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), no. 1, 1–34. MR 1038356, DOI 10.2307/1971506
- David Hoffman and William H. Meeks III, Minimal surfaces based on the catenoid, Amer. Math. Monthly 97 (1990), no. 8, 702–730. MR 1072813, DOI 10.2307/2324576 D. Hoffman and M. Wohlgemuth, Limiting behavior of classical periodic minimal surfaces, GANG preprint series III (to appear). J. T. Hoffman, MESH manual, GANG preprint series II, #35.
- H. Karcher, Embedded minimal surfaces derived from Scherk’s examples, Manuscripta Math. 62 (1988), no. 1, 83–114. MR 958255, DOI 10.1007/BF01258269
- Francisco J. López and Antonio Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), no. 1, 293–300. MR 1085145, DOI 10.4310/jdg/1214446040
- William H. Meeks III and Harold Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989), no. 2, 351–379. MR 1001845, DOI 10.1007/BF01389046
- Johannes C. C. Nitsche, Lectures on minimal surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems; Translated from the German by Jerry M. Feinberg; With a German foreword. MR 1015936
- Robert Osserman, Global properties of minimal surfaces in $E^{3}$ and $E^{n}$, Ann. of Math. (2) 80 (1964), 340–364. MR 179701, DOI 10.2307/1970396
- Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409 I. Peterson, Three bites in a doughnut, Sci. News 127 (1985), 168-169. H. F. Scherk, Bemerkungen über die kleinste fläche innerhalb gegebener grenzen, J. Reine Angew. Math. 13 (1835), 185-208.
- Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928, DOI 10.4310/jdg/1214438183
- Fu Sheng Wei, Some existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math. 109 (1992), no. 1, 113–136. MR 1168368, DOI 10.1007/BF01232021
- Meinhard Wohlgemuth, Higher genus minimal surfaces by growing handles out of a catenoid, Manuscripta Math. 70 (1991), no. 4, 397–428. MR 1092145, DOI 10.1007/BF02568387
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 29 (1993), 77-84
- MSC (2000): Primary 53A10; Secondary 58E12
- DOI: https://doi.org/10.1090/S0273-0979-1993-00401-4
- MathSciNet review: 1193537