## The closure of spectral data for constant mean curvature tori in $$S^3$$

### Emma Carberry and Martin Ulrich Schmidt

#### 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 $$n$$.

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 Wednesday, February 22, 2012