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

TL;DR

This paper proves that spectral curves of constant mean curvature tori in the 3-sphere are dense among all finite-type solutions, leading to numerous families of such tori with varying dimensions.

Contribution

It demonstrates the density of spectral curves of CMC tori in the space of all finite-type spectral curves, revealing a rich structure of CMC tori families.

Findings

Spectral curves of CMC tori are dense in the space of finite-type solutions.

Existence of countably many real n-dimensional families of CMC tori for each positive integer n.

The spectral curve correspondence characterizes CMC tori via hyperelliptic curves with real structures.

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 $.