Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

摘要

Given an oriented Riemannian surface (σ,g), its tangent bundle Tσ enjoys a natural pseudo-K?hler structure, that is the combination of a complex structure J, a pseudo-metric G with neutral signature and a symplectic structure Ω. We give a local classification of those surfaces of Tσ which are both Lagrangian with respect to Ω 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_1~3 induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS~2 or TH~2 respectively. We relate the area of the congruence to a second-order functional F=∫√H~2-KdA on the original surface.