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

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 $n$.

20 pages

References in corpus (2)

• The geometric complexity of special Lagrangian $T^2$-cones
• On infinitesimal deformations of cmc surfaces of finite type in the 3-sphere

Cited by in corpus (4)

• The Prevalence of Tori amongst Constant Mean Curvature Planes in ${\mathbb{R}}^3$
• Whitham Deformations and the Space of Harmonic Tori in ${\mathbb{S}}^3$
• Harmonic tori in de Sitter spaces $S_1^{2n}$
• The space of equivariant harmonic tori in the 3-sphere