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Constant mean curvature tori in terms of elliptic functions. (English) Zbl 0597.53003

The purpose of this paper is to give an explicit description of the constant mean curvature tori in \({\mathbb{R}}^ 3\) discovered by H. C. Wente [Pac. J. Math. 121, 193-243 (1986; Zbl 0586.53003)]. In fact, we classify all constant mean curvature tori that have one family of planar curvature lines.
Classical results in differential geometry reduce the problem to finding special doubly periodic solutions of the sinh-Gordon equation \(\Delta \omega +\sinh \omega \cdot \cosh \omega =0\) in the plane; the extra conditions arise, since finitely many of the congruent pieces corresponding to a fundamental domain of \(\omega\) and its orbit in \({\mathbb{R}}^ 2\) must fit together to form a closed surface. The condition on the torsion of the curvature lines, which was suggested by a numerical investigation of Wente’s approach, yields an additional partial differential equation for \(\omega\), altogether an overdetermined system. This system is integrable, and the common solutions \(\omega\) are given in terms of ordinary differential equations; they can be integrated using Jacobi elliptic functions.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35N10 Overdetermined systems of PDEs with variable coefficients

Citations:

Zbl 0586.53003
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