Minimal surfaces under constrained Willmore transformation

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Abstract

The class of constrained Willmore (CW) surfaces in space-forms constitutes a Möbius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation [8], defined by the action of a loop of flat metric connections, and Bäcklund transformations [9], defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are [25] examples of CW surfaces, characterized by the existence of some polynomial conserved quantity [21, 22, 24]. Both CW spectral deformation and CW Bäcklund transformation preserve [21, 22, 24] the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore [21, 22, 24], preservation of both the space-form and the mean curvature, in the latter case. A classical result by Thomsen [28] characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves [8, 9] the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation [8]. We define, in this way, a CW spectral deformation and CW Bäcklund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.

Original languageEnglish
Title of host publicationMinimal Surfaces
Subtitle of host publicationIntegrable Systems and Visualisation - Workshops, 2016-19
EditorsTim Hoffmann, Martin Kilian, Katrin Leschke, Francisco Martin
PublisherSpringer
Pages229-245
Number of pages17
ISBN (Print)9783030685409
DOIs
Publication statusPublished - 2021
EventWorkshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19 - Cork, Ireland
Duration: 27 Mar 201729 Mar 2017

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume349
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceWorkshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19
CountryIreland
CityCork
Period27/03/1729/03/17

Keywords

  • Bäcklund transformations
  • Constant mean curvature surfaces
  • Constrained Willmore surfaces
  • Isothermic surfaces
  • Minimal surfaces
  • Polynomial conserved quantities
  • Spectral deformation
  • Willmore energy

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