The spectral data for Hamiltonian stationary Lagrangian tori in R^4

Abstract : Hamiltonian stationary Lagrangian submanifolds are solutions of a natural and important variational problem in Kaehler geometry. In the particular case of surfaces in Euclidean 4-space, it has recently been proved that the Euler-Lagrange equation is a completely integrable system, which theory allows us to describe all such tori. This article determines the spectral data for these, in terms of a complete algebraic curve, a rational function and a line bundle. We use this data to give explicit formulas for all weakly conformal HSL immersions of a 2-torus into Euclidean 4-space and describe the moduli space of those with given conformal type and Maslov class. We also show that each such torus admits a family of Hamiltonian deformations through HSL tori, the dimension of this family being related to the genus of its spectral curve.
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https://hal.archives-ouvertes.fr/hal-00163082
Contributor : Pascal Romon <>
Submitted on : Monday, July 16, 2007 - 4:41:10 PM
Last modification on : Wednesday, September 4, 2019 - 1:52:03 PM

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Ian Mcintosh, Pascal Romon. The spectral data for Hamiltonian stationary Lagrangian tori in R^4. Differential Geometry and its Applications, Elsevier, 2011, 29 (2), pp.125-146. ⟨10.1016/j.difgeo.2011.02.007⟩. ⟨hal-00163082⟩

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