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Computational aspects of soap bubble deformations. (English) Zbl 0799.53007
The authors search for the simplest possible one-parameter family of constant mean curvature (CMC) tori; deformations presented are given by a closed one-dimensional orbit in the moduli space and are distinguished by having one family of spherical curvature lines. Such tori have been studied by Wente [H. C. Wente, Mem. Am. Math. Soc. 478, 77 p. (1992; Zbl 0772.53004)] based on a method due to Dobriner [H. Dobriner, Acta Math. 9, 73-104 (1886)]. In a previous paper the authors have proved several results on the CMC tori [U. Pinkall and I. Sterling, Ann. Math., II. Ser. 130, No. 2, 407-451 (1989; Zbl 0683.53053)]. In this new paper they give two ways to compute the one- dimensional orbit of CMC tori corresponding to one that has been found. They then use their computing methods to find seven-lobed tori and deforming soliton-like cylinders with embedded Delaunay ends. The paper contains several illustrations of computing results.
Reviewer: M.Emmer (Roma)

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Hermann Dobriner, Die flächen constanter krümmung mit einem system sphärischer krümmungslinien dargestellt mit hilfe von thetafunctionen zweier variabeln, Acta Math. 9 (1887), no. 1, 73 – 104 (German). · JFM 18.0425.01 · doi:10.1007/BF02406731 · doi.org
[2] U. Pinkall and I. Sterling, On the classification of constant mean curvature tori, Ann. of Math. (2) 130 (1989), no. 2, 407 – 451. · Zbl 0683.53053 · doi:10.2307/1971425 · doi.org
[3] H. Wente, Constant mean curvature immersions with one family of curvature lines spherical, preprint No. 10, SFB 256, Bonn, 1988.
[4] Henry C. Wente, Constant mean curvature immersions of Enneper type, Mem. Amer. Math. Soc. 100 (1992), no. 478, vi+77. · Zbl 0772.53004 · doi:10.1090/memo/0478 · doi.org
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