Minimal Tori in $S^3$

TL;DR

This paper establishes the existence of multiple families of minimal immersions of 2-tori into the 3-sphere, using algebraic curve data and periodicity conditions to classify these immersions.

Contribution

It proves the existence of countably many families of minimal tori in $S^3$ and characterizes the space of such immersions using Hitchin's correspondence.

Findings

Countably many families of minimal tori in $S^3$ exist.

Every linearly full minimal torus belongs to a unique family.

A dense subset of rectangular tori can be minimally immersed into $S^3$.

Abstract

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.