Constrained Willmore Surfaces: Symmetries of a Moebius Invariant Integrable System

TL;DR

This paper investigates the symmetries and transformations of constrained Willmore surfaces, a class of surfaces invariant under Moebius transformations, introducing new deformation and transformation techniques and analyzing their properties.

Contribution

It defines spectral, Baecklund, and Darboux transformations for constrained Willmore surfaces, establishing their relationships and invariance properties, especially in 4-space.

Findings

Spectral deformation and Baecklund transformation commute.

Darboux transformations are special cases of Baecklund transformations in 4-space.

Transformations preserve the class of constrained Willmore surfaces with conserved quantities.

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.