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Article Dans Une Revue Journal of Geometry and Physics Année : 2011

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Résumé

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $T\Sigma$ which are both Lagrangian with respect to $\Om$ and minimal with respect to $\G$. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\R^3$ or $\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T\S^2$ or $T \H^2$ respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.

Dates et versions

hal-00333742 , version 1 (24-10-2008)

Identifiants

Citer

Henri Anciaux, Brendan Guilfoyle, Pascal Romon. Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface. Journal of Geometry and Physics, 2011, 61 (1), pp.237-247. ⟨10.1016/j.geomphys.2010.09.017⟩. ⟨hal-00333742⟩
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