Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space

TL;DR

This paper explores the connection between integrable systems and minimal surface theory by demonstrating that the Lopez-Ros deformation is a special case of simple factor dressing, enabling new constructions of minimal surfaces.

Contribution

It introduces a new perspective linking dressing transformations to classical minimal surface deformations, providing explicit formulas and control over surface properties.

Findings

Lopez-Ros deformation is a special case of simple factor dressing.

Explicit formulas for dressing and Lopez-Ros deformation in terms of minimal surfaces.

Construction of new doubly-periodic minimal surfaces from Scherk's surface.

Abstract

The aim of this paper is to give 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 Lopez-Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the Lopez-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.