Constrained Willmore Surfaces: Symmetries of a Moebius Invariant Integrable System

Abstract

This work is dedicated to the study of the Moebius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Baecklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation. We establish a permutability between spectral deformation and Baecklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Baecklund transformation. All these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We verify that, for special choices of parameters, both spectral deformation and Baecklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of CMC surfaces in 3-dimensional space-form.

Based on the author's PhD Thesis, submitted to the University of Bath, UK on September 2008; 225 pages; 2 figures. v2: January 2010 errata to the printed version attached to v1