The classification of doubly periodic minimal tori with parallel ends
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
Perez, Joaquin et al.
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2005