Minimal Tori in $S^3$

Details Minimal Tori in $S^3$ Minimal Tori in $S^3$

2004-07-16

arxiv.org/abs/math/0407304v2

Pacific Journal of mathematics, Nov 2007, vol 233, no 1, pp41-70

Mathematics

Differential Geometry

To appear, Pacific Journal of mathematics. 27 pages, 7 figures. Minor changes only

Scientific paper

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.

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