Annals of Global Analysis and Geometry

, Volume 43, Issue 4, pp 299–329 | Cite as

Darboux transforms and spectral curves of constant mean curvature surfaces revisited

Article

Abstract

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach.

Keywords

Constant mean curvature surfaces Harmonic maps Integrable systems 

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  2. 2.Department of MathematicsUniversity of LeicesterLeicesterUK
  3. 3.Mathematisches InstitutTübingenGermany
  4. 4.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA

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