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arXiv · 1201.0190

Transformations of harmonic bundles and Willmore surfaces

A. C. Quintino
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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.

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A. C. Quintino. 2011-12-30. Transformations of harmonic bundles and Willmore surfaces. https://arxiv.org/abs/1201.0190

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