Minimal Tori in $S^3$

We prove existence results that give information about the space of minimal immersions of 2-tori into $S ^ 3$. More specifically, we show that \begin{enumerate} \item For every positive integer $n$, there are countably many real $n$-dimensional families of minimally immersed 2-tori in $S ^ 3$. Every linearly full minimal immersion $T ^ 2\to S ^ 3$ belongs to exactly one of these families. \item Let $\mathcal A$ be the space of rectangular 2-tori. There is a countable dense subset $\mathcal B$ of $\mathcal A$ such that every torus in $\mathcal B$ can be minimally immersed into $S ^ 3$. \end{enumerate} The main content of this manuscript lies in finding minimal immersions that satisfy {\bf periodicity conditions} and hence obtaining maps of tori, rather than simply immersions of the plane. We make use of a correspondence, established by Hitchin, between minimal tori in $S^3$ and algebraic curve data.