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

In Pinkall and Sterling (Ann Math 130:407–451, 1989) and Hitchin (J Differ Geom 31:627–710, 1990) the solutions of the elliptic sinh-Gordon equation of finite spectral genus g are investigated. 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. For \(g=2\) $g=2$ they are double periodic. The eigenvalues of these matrix-valued polynomials are preserved along the flows and determine the lattice of periods. For \(g = 2\) $g=2$ , 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. These limiting cases are used to construct on every elliptic curve three conformal maps to \(\mathbb {H}\) $\mathbb{H}$ which are constrained Willmore. Finally, the Willmore functional is calculated in dependence of the conformal class.