The classification of doubly periodic minimal tori with parallel ends. (English) Zbl 1094.53007

Let \(\kappa\) be the space of properly embedded minimal tori in a quotient of \(\mathbb R^3\) by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, the authors prove that \(\kappa \) is a \(3\)-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by H. Karcher, W. H. Meeks and H. Rosenberg. The degenerate limits of surfaces in \(\kappa\) are the catenoid, the helicoid and three \(1\)-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples.


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
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