The closure of spectral data for constant mean curvature tori in 𝕊3

Abstract

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 ${\mathbb{S}^{3}}$ 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 ${\mathbb{S}^{3}}$ for each positive integer n.