The classification of doubly periodic minimal tori with parallel ends

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