Vol. 233, No. 1, 2007

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Mininal tori in S3

Emma Carberry

Vol. 233 (2007), No. 1, 41–69

We prove existence results that give information about the space of minimal immersions of 2-tori into S3. More specifically, we show:

  • For every positive integer n, there are countably many real n-dimensional families of minimally immersed 2-tori in S3. Every linearly full minimal immersion T2 S3 belongs to exactly one of these families.
  • Let 𝒜 be the space of rectangular 2-tori. There is a countable dense subset of 𝒜 such that every torus in can be minimally immersed into S3.

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 S3 and algebraic curve data.

minimal surfaces, integrable systems, spectral curves
Mathematical Subject Classification 2000
Primary: 53C42
Secondary: 14H70
Received: 13 September 2006
Accepted: 16 February 2007
Published: 1 November 2007
Emma Carberry
Department of Mathematics and Statistics
Carslaw Building, F07
University of Sydney
NSW, 2006