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Darboux transforms and spectral curves of constant mean curvature surfaces revisited

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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.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    E. Carberry

  2. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK

    K. Leschke

  3. Mathematisches Institut, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    F. Pedit

  4. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 01003-9305, USA

    F. Pedit

Authors
  1. E. Carberry
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  2. K. Leschke
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  3. F. Pedit
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Correspondence to E. Carberry.

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Carberry, E., Leschke, K. & Pedit, F. Darboux transforms and spectral curves of constant mean curvature surfaces revisited. Ann Glob Anal Geom 43, 299–329 (2013). https://doi.org/10.1007/s10455-012-9347-8

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