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On the classification of constant mean curvature tori. (English) Zbl 0683.53053

The study of constant mean curvature (cmc) surfaces without umbilic points in 3-space is reduced to solving the Gauss equation, which after suitable normalizations becomes the elliptic sinh-Gordon equation. Using infinitesimal deformations (“Jacobi fields”) the authors construct an infinite series of solutions to the linearized sinh-Gordon equation which in the torus case become linearly dependent, and therefore reduce the problem to an ODE system on a jet space. Consideration of the associated family of cmc surfaces produces an embedding of the ODE system into the Jacobian of a certain hyperelliptic Riemann surface. Here the flow becomes linear, and the question of closedness (periodicity) can be effectively solved. The dimension 6n-1 of the ODE characterizes “type n” solutions, and the lowest values correspond to the famous Wente and Abresch cmc tori. The paper contains comments on a computer program for the search of closed examples using Newton’s method. Several pictures of new cmc tori are given.
Reviewer: D.Ferus

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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