Abstract
The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.
Highlights
Minimal surfaces, that is, surfaces with vanishing mean curvature, first implicitly appeared as solutions to the Euler-Lagrange equation of the area functional in [42] by Lagrange
In the case of a minimal surface, we show that the harmonic maps in the associated family of the conformal Gauss map are the conformal Gauss maps of the associated family of minimal surfaces
For the most simple dressing operation given by a dressing matrix with a simple pole, the so-called simple factor dressing, the new harmonic map can be computed explicitly and is the conformal Gauss map of a new Willmore surface in the 4-sphere [4,43]
Summary
That is, surfaces with vanishing mean curvature, first implicitly appeared as solutions to the Euler-Lagrange equation of the area functional in [42] by Lagrange. We will briefly recall the construction of the associated families dλ and dλS of flat connections for both the harmonic Gauss map N and the conformal Gauss map S of a minimal surface in our setup Both are closely related: parallel sections of dλS can be expressed in terms of parallel sections of dλ and generalisations f p,q , p, q ∈ S3, of the associated family of minimal surfaces fcos θ,sin θ. For the most simple dressing operation given by a dressing matrix with a simple pole, the so-called simple factor dressing, the new harmonic map can be computed explicitly and is the conformal Gauss map of a new Willmore surface in the 4-sphere [4,43]. The first author would like to thank the members of both institutions for their hospitality during her stay, and the University of Leicester for granting her study leave
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