Solutions of the Sinh-Gordon Equation of Spectral Genus Two and constrained Willmore Tori I

TL;DR

This paper studies solutions to the elliptic sinh-Gordon equation of spectral genus less than 3, exploring their spectral properties, periodicity, and applications to constructing constrained Willmore tori in four-dimensional space.

Contribution

It provides a detailed analysis of spectral genus two solutions, their isospectral sets, and constructs constrained Willmore tori on elliptic curves, extending understanding of these geometric structures.

Findings

Eigenvalues determine period lattices of solutions.

Constructed three constrained Willmore tori on elliptic curves.

Calculated Willmore functional dependence on conformal class.

Abstract

We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double periodic. The eigenvalues of these matrix-valued polynomials are preserved along the flows and determine the lattice of periods. We investigate the level sets of these eigenvalues, which are called isospectral sets, and the dependence of the period lattice on the isospectral sets. The limiting cases of spectral genus one and zero are included. Moreover, these limiting cases are used to construct on every elliptic curve three conformal maps to R^4 which are constrained Willmore. Finally, the Willmore functional is calculated in dependence of the conformal class.