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The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. (English) Zbl 0687.53010
The existence of certain triply periodic minimal surfaces in $${\mathbb{R}}^ 3$$ previously obtained by A. Schoen (and of more such surfaces) is proved by considering conjugate polygonal Plateau problems. Deformations of these to triply periodic constant mean curvature surfaces are obtained by solving a conjugate Plateau problem in $$S^ 3$$. The corresponding polygons in $$S^ 3$$ have edges on parallel geodesics where the Euclidean polygons have parallel edges. Also new compact embedded minimal surfaces in $$S^ 3$$ are obtained by this method. For many of the Euclidean minimal surfaces explicit Weierstrass representations are derived.
Reviewer: U.Pinkall

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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##### References:
 [1] Schoen, A.H.: Infinite periodic minimal surfaces without selfintersections. NASA Technical Note No. TN D-5541 (1970) · Zbl 1071.53507 [2] Anderson, D.: Studies in the Microstructure of Microemulsions. PhD thesis, University of Minnesota, June 1986 [3] Andersson, S., Hyde, S.T., Larsson, K., Lidin, S.: Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers. Chemical Reviews88, 221-242 (1988) · doi:10.1021/cr00083a011 [4] Fischer, W., Koch, E.: On 3-periodic minimal surfaces. Zeitschrift für Kristallographie179, 31-52 (1987) and Galleys 1988 · Zbl 0684.53010 · doi:10.1524/zkri.1987.179.1-4.31 [5] Hyde, S.: Infinite periodic minimal surfaces and crystal structures. PhD thesis, Dept. of Physics, Monash University, March 1986 [6] Darboux, G.: Théorie Générale des Surface I. Gauthier-Villars, Paris 1887 [7] Karcher, H., Pinkall, U., Sterling, I.: New Minimal Surfaces in S3. Preprint 86-27 Max-Planck-Institut f. Mathematik, Bonn. Submitted JDG 1986 · Zbl 0653.53004 [8] Lawson, B.H.: Complete Minimal Surfaces in S3. Annals of Math.92, 335-374 (1970) · Zbl 0205.52001 · doi:10.2307/1970625 [9] Nitsche, J.C.C.: Über ein verallgemeinertes Dirichletsches Problem für die Minimalflächengleichung und hebbare Unstetigkeiten ihrer Lösungen. Math. Ann.158, 203-214 (1965) · Zbl 0141.09601 · doi:10.1007/BF01360040 [10] Neovius, E.R.: Bestimmung Zweier Specieller Periodischer Minimalflächen. Helsingfors 1883 [11] Nagano, T., Smyth, B.: Periodic Minimal Surfaces and Weyl Groups. Acta Math.145, 1-27 (1980) · Zbl 0449.53042 · doi:10.1007/BF02414183 [12] Osserman, R.: Global Properties of Minimal Surfaces in E3 and En. Ann. of Math.80, 340-364 (1964) · Zbl 0134.38502 · doi:10.2307/1970396 [13] Schwarz, H.A.: Gesammelte Mathematische Abhandlungen. Springer, Berlin 1890 [14] Smyth, B.: Stationary minimal surfaces with boundary on a simplex. Invent. Math.76, 411-420 (1984) · Zbl 0538.53010 · doi:10.1007/BF01388467 [15] Wohlgemuth, M.: Abelsche Minimalflächen. Diplomarbeit, Bonn 1988
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