Transformations of harmonic bundles and Willmore surfaces

TL;DR

This paper explores the transformations and deformations of Willmore surfaces through harmonic bundle theory, providing a conformally-invariant framework and new insights into their geometric properties.

Contribution

It offers a self-contained, conformally-invariant account of harmonic bundle transformations and their application to Willmore surface deformations.

Findings

Establishes a zero-curvature formulation for Willmore surfaces.

Defines spectral and Bäcklund transformations for Willmore surfaces.

Demonstrates Bianchi permutability between transformations.

Abstract

Willmore surfaces are the extremals of the Willmore functional (possibly under a constraint on the conformal structure). With the characterization of Willmore surfaces by the (possibly perturbed) harmonicity of the mean curvature sphere congruence [Blaschke, Ejiri, Rigoli, Burstall-Calderbank], a zero-curvature formulation follows [Burstall-Calderbank]. Deformations on the level of harmonic maps prove to give rise to deformations on the level of surfaces, with the definition of a spectral deformation [Burstall-Pedit-Pinkall, Burstall-Calderbank] and of a Baecklund transformation [Burstall-Quintino] of Willmore surfaces into new ones, with a Bianchi permutability between the two [Burstall-Quintino]. This text is dedicated to a self-contained account of the topic, from a conformally-invariant viewpoint, in Darboux's light-cone model of the conformal $n$-sphere.