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Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation

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Abstract

We study 3-dimensional minimal Lagrangian submanifolds of the 3-dimensional complex projective space ℂP3 (4) which admit a unit length Killing vector field whose integral curves are geodesics. We show that such Lagrangian submanifolds can be obtained from either horizontal holomorphic curves in ℝP3 (4) (or equivalently superminimal immersions of surfaces in S4 (1)) or from solutions of the two dimensional sinh-Gordon equation. In the latter case, we explicitly obtain the immersions in terms of elliptic functions in the case that the solutions of the sinh-Gordon equation depend only on one variable.

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Authors and Affiliations

  1. Departamento de Matemáticas, Escuela Politécnica Superior Universidad de Jaén, 23071, Jaén, Spain

    Ildefonso Castro

  2. Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, 3584CD, Utrecht, The Netherlands

    Luc Vrancken

Authors
  1. Ildefonso Castro
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  2. Luc Vrancken
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Correspondence to Ildefonso Castro.

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Dedicated to S.S. Chern on the occasion of his 90th birthday

Partially supported by a research fellowship of the Alexander von Humboldt Stiftung (Germany)

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Castro, I., Vrancken, L. Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation. Results. Math. 40, 130–143 (2001). https://doi.org/10.1007/BF03322703

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  • DOI: https://doi.org/10.1007/BF03322703

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