The space of genus two spectral curves of constant mean curvature tori in $\mathbb{R}^3$

TL;DR

This paper characterizes the spectral data of genus two constant mean curvature tori in three-dimensional space using Whitham deformations, revealing a geometric parameterization and analyzing boundary behaviors.

Contribution

It provides a complete parameterization of spectral data for genus two CMC tori via an isosceles right triangle and introduces innovative methods combining blowups and spectral data analysis.

Findings

Spectral data of genus two CMC tori form a geometric triangle parameter space.

The Wente family corresponds to the bisector of the right angle in this parameter space.

Boundary analysis reveals the structure of spectral data limits for these tori.

Abstract

We use Whitham deformations to give a complete account of spectral data of real solutions of the sinh--Gordon equation of spectral genus 2. We parameterise the closure of spectral data of constant mean curvature tori in $\mathbb{R}^3$ by an isosceles right triangle and analyse its boundary. We prove that the Wente family, which is described by spectral data with real coefficients, is parameterised by the bisector of the right angle. Our methods combine blowups of Whitham deformations and spectral data in an innovative way that changes the underlying integrable system.