The space of doubly periodic minimal tori with parallel ends: Standard Examples

We describe a 3-parametric family $\mathcal{K}$ of properly embedded minimal tori with four parallel ends in quotients of $\mathbb{R}^3$ by two independent translations, which we will call the \textit{Standard Examples.} These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. $\mathcal{K}$ can be endowed with a natural structure of a self-conjugated 3-dimensional real analytic manifold diffeomorphic to $\mathbb{R}\times(\mathbb{R}^2-\{(\pm 1,0)\})$ whose degenerate limits are the catenoid, the helicoid, the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples. Perez, Rodriguez and Traizet \cite{PeRoTra1} characterize $\mathcal{K}$ in the following sense: If $M$ is a properly embedded minimal torus in a quotient of $\mathbb{R}^3$ by two independent translations with any number of parallel ends, then $M$ is a finite covering of a standard example.