Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Abstract

Given an oriented Riemannian surface $(\Sigma ,g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-Kähler 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 ${\mathbb{R}}^3$ or ${\mathbb{R}}_1^3$ 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.

22 pages, typos corrected, results streamlined

References in corpus (1)

• On Indefinite Special Lagrangian Submanifolds in Indefinite Complex Euclidean Spaces

Cited by in corpus (6)

• On minimal Lagrangian surfaces in the product of Riemannian two manifolds
• A new geometric structure on tangent bundles
• Marginally trapped surfaces in spaces of oriented geodesics
• On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms
• A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold
• Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces