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arXiv · 1202.4517

The Closure of Spectral Data for Constant Mean Curvature Tori in $ S ^ 3 $

Emma Carberry·Martin Ulrich Schmidt
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Abstract

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $ S ^ 3 $ result when these spectral curves satisfy periodicity conditions. We prove that the spectral curves of CMC tori are dense in the space of smooth spectral curves of finite-type solutions of the sinh-Gordon equation. One consequence of this is the existence of countably many real $ n $-dimensional families of CMC tori in $ S ^ 3 $ for each positive integer $ n $.

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Emma Carberry, Martin Ulrich Schmidt. 2012-02-21. The Closure of Spectral Data for Constant Mean Curvature Tori in $ S ^ 3 $. https://doi.org/10.1515/crelle-2014-0068

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