Spectral deformation and B?cklund transformation of constrained Willmore surfaces

摘要

The class of constrained Willmore surfaces in space-forms forms a M?bius invariant class of surfaces with strong links to the theory of integrable systems. This paper is dedicated to an overview on the topic. We define a spectral deformation, by the action of a loop of flat metric connections, and B?cklund transformations, by applying a dressing action. We establish a permutability between spectral deformation and B?cklund transformation and verify that all these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We show that, for special choices of parameters, both spectral deformation and B?cklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of constant mean curvature surfaces in 3-dimensional space-forms.