TY - JOUR

T1 - The spectral data for Hamiltonian stationary Lagrangian tori in R^4

AU - McIntosh, Ian

AU - Romon, Pascal

PY - 2011/3/1

Y1 - 2011/3/1

N2 - 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.

AB - 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.

KW - Geometry

UR - http://www.scopus.com/inward/record.url?scp=79952487544&partnerID=8YFLogxK

U2 - 10.1016/j.difgeo.2011.02.007

DO - 10.1016/j.difgeo.2011.02.007

M3 - Article

VL - 29

SP - 125

EP - 146

JO - Differential Geometry and its Applications

JF - Differential Geometry and its Applications

IS - 2

ER -