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

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posted on 2019-02-12, 16:38 authored by K. Leschke, K. Moriya
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.

Funding

K. Leschke was partially supported by DFG SPP 1154 “Global Differential Geometry” and JSPS KAKENHI Grant-in-Aids for Scientific Research (C), Grant number 24540090. K. Leschke and K. Moriya were supported by JSPS KAKENHI Grant-in-Aids for Scientific Research (C), Grant numbers 22540064 and 25400063.

History

Citation

Mathematische Zeitschrift, 2018

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • VoR (Version of Record)

Published in

Mathematische Zeitschrift

Publisher

Springer (part of Springer Nature)

issn

0025-5874

eissn

1432-1823

Acceptance date

2018-12-04

Copyright date

2018

Available date

2019-02-12

Publisher version

https://link.springer.com/article/10.1007/s00209-018-2217-8

Language

en

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