Dressing transformations of constrained Willmore surfaces

TL;DR

This paper develops a method to generate transformations of constrained Willmore surfaces in any codimension, extending classical transforms and proving their permutability, with special focus on codimension 2 cases.

Contribution

It adapts the dressing method and Bäcklund transforms to constrained Willmore surfaces, generalizing existing transforms and establishing Bianchi permutability.

Findings

Constructed transformations using dressing method.

Proved Bianchi permutability of these transformations.

Generalized Darboux transforms to constrained Willmore surfaces.

Abstract

We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define B\"acklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to codimension 2, we generalise the Darboux transforms of Willmore surfaces via Riccati equations, due to Burstall-Ferus-Leschke-Pedit-Pinkall, to the constrained Willmore case and show that they amount to our B\"acklund transforms with real spectral parameter.