The classification of doubly periodic minimal tori with parallel ends

Details The classification of doubly periodic minimal tori with parallel ends The classification of doubly periodic minimal tori with parallel ends

2005-01-28

arxiv.org/abs/math/0501507v1

Mathematics

Differential Geometry

55 pages, 8 figures

Scientific paper

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. The degenerate limits of surfaces in $\mathcal{K}$ 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.

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