Mininal tori in S³

Abstract

We prove existence results that give information about the space of minimal immersions of 2-tori into S 3 . More specifically, we show:• 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 → S 3 belongs to exactly one of these families.• Let Ꮽ be the space of rectangular 2-tori. There is a countable dense subset Ꮾ of Ꮽ such that every torus in Ꮾ can be minimally immersed into S 3 .Mainly, we find minimal immersions that satisfy periodicity conditions and hence obtain maps of tori, rather than simply immersions of the plane. This work uses a correspondence, established by Hitchin, between minimal tori in S 3 and algebraic curve data.

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“…Mean curvature and minimal tori. There are infinitely many minimal tori in S 3 [5]. There are in fact already infinitely many minimal equivariant ones [12].…”

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“…Theorem 2 (Carberry, [5]). For each integer n 0, there are countably many real n-dimensional families of minimal immersions from rectangular tori to S 3 .…”

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