A rank 2 theory for constrained Willmore tori in the 3-dimensional sphere

Abstract

The main subject of this thesis are constrained Willmore tori in the 3-dimensional sphere S^3. It is known that constrained Willmore tori in the 4-sphere come with an associated C_∗-family of flat SL(4,C)-connections. This allows to study constrained Willmore tori as an integrable system. The initial surface can be reconstructed by holomorphic data on the spectralcurve Σ, which is the riemann surface on which the eigenlines of the family of connections are well-defined. If the constrained Willmore torus lies in a 3-dimensional sphere, there is a further symmetry on the spectral curve, a holomorphic involution σ. In this thesis we show that this involution allows to reduce the family of SL(4,C)-connections into a family of SL(2,C)-connections. We achieve this by pushing forward the eigenline bundle of the rank 4 family on the quotient surface Σ/σ. Therefore,the rank 2 family of connections is a Σ/σ-family of connections, which is a hyperelliptic surface. The rank 2 family of connections then allows to give a Sym-Bobenko formula, similiar to the case of constant mean curvature surfaces in S^3. Further, if the quotient surface Σ/σ=CP1, the surface is of constant mean curvature in a space form.