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The computer-aided discovery of new embedded minimal surfaces. (English) Zbl 0616.53007
This is a report on some recent developments in the theory of minimal surfaces. After some historical remarks the construction of the author’s and B. Meeks’ examples of complete embedded minimal surfaces of finite topological type is described, beginning with C. Costa’s example from 1984. The use of computer graphics for the construction of the examples is exhibited. The mathematical background has been published in several recent papers.
Reviewer: Bernd Wegner

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
Full Text: DOI
[1] C. Costa, ”Imersões minimas completas em R3 de gênero um e curvatura total finita,” Doctoral thesis, IMPA, Rio de Janeiro, Brasil, 1982. (Example of a complete minimal immersion in R3 of genus one and three embedded ends.Bull. Soc. Bras. Mat., 15 [1984] 47–54.) · Zbl 0613.53002 · doi:10.1007/BF02584707
[2] L. Barbosa and G. Colares,Examples of Minimal Surfaces, to appear in Springer Lecture Notes series. · Zbl 0609.53001
[3] J-P. Bourguignon, H. B. Lawson and C. Margenn, ”Les surfaces minimales,”Pour la Science, January 1986.
[4] S. Hildebrandt and A. J. Tromba,The Mathematics of Optimal Form, Scientific American Library, 1985, W. H. Freeman, New York. · Zbl 0860.01004
[5] D. Hoffman,The discovery of new embedded minimal surfaces: elliptic functions; symmetry; computer graphics, Proceedings of the Berlin Conference on Global Differential Geometry, Berlin, June 1984.
[6] D. Hoffman,The construction of families of embedded minimal surfaces, to appear in the Proceddings of the Stanford Conference on Variational Methods for Free Surface Interfaces, September 1985.
[7] D. Hoffman and W. Meeks III, ”Complete embedded minimal surfaces of finite total curvature,”Bull. A.M.S., January 1985, 134-136. · Zbl 0566.53017
[8] D. Hoffman and W. Meeks III, ”A complete embedded minimal surface with genus one, three ends and finite total curvature,”Journal of Differential Geometry, March 1985. · Zbl 0604.53002
[9] D. Hoffman and W. Meeks III,The global theory of embedded minimal surfaces, to appear. · Zbl 0634.53003
[10] L. Jorge and W. Meeks III, ”The topology of complete minimal surfaces of finite total Gaussian curvature,”Topology, 22 No. 2, 1983, 203–221. · Zbl 0517.53008 · doi:10.1016/0040-9383(83)90032-0
[11] H. B. Lawson Jr,Lectures on Minimal Submanifolds, Publish or Perish Press, Berkeley, 1971.
[12] J. C. C. Nitsche,Minimal surfaces and partial differential equations, inStudies in Partial Differential Equations, Walter Littman, ed., MAA Studies in Mathematics, Vol. 23, Mathematical Association of America, 1982. · Zbl 0533.53003
[13] R. Osserman, ”Global properties of minimal surfaces in E3 and En,”Ann. of Math., 80 (1984), 340–364. · Zbl 0134.38502 · doi:10.2307/1970396
[14] R. Osserman,A Survey of Minimal Surfaces, 2d Edition, Dover Publications, New York, 1986. · Zbl 0209.52901
[15] R. Schoen, ”Uniqueness, symmetry, and embeddedness of minimal surfaces,”J. Dif. Geom. 18, 1983, 791–809. · Zbl 0575.53037
[16] I. Peterson, ”Three Bites in a Doughnut,”Science News 127, No. 11, 16 March 1985.
[17] Dr. Crypton, ”Shapes that eluded discovery,”Science Digest, April 1986.
[18] S. Papert,Mindstorms; Children, Computers, and Powerful Ideas, Basic Books, New York, 1980.
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