Dressing transformations of 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 Bäcklund 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 Bäcklund transforms with real spectral parameter.

31 A4 pages. v2: metadata adjusted. v3: references added; minor clarifications made

References in corpus (2)

• Conformal submanifold geometry I-III
• Harmonic map methods for Willmore surfaces

Cited by in corpus (7)

• Notes on transformations in integrable geometry
• On the Björling problem for Willmore surfaces
• Construction of Willmore two-spheres via harmonic maps into $S{O}^{+}(1,n+3)/(S{O}^{+}(1,1)\times SO(n+2))$
• Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space
• Polynomial conserved quantities for constrained Willmore surfaces
• Quartic differentials and harmonic maps in conformal surface geometry
• Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$