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 $n$.