Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface (Σ,g)(Σ,g), its tangent bundle TΣTΣ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure JJ, a pseudo-metric GG with neutral signature and a symplectic structure ΩΩ. We give a local classification of those surfaces of TΣTΣ which are both Lagrangian with respect to ΩΩ and minimal with respect to GG. We first show that if gg 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 R3R3 or R13 induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS2TS2 or TH2TH2 respectively. We relate the area of the congruence to a second-order functional F=∫H2−KdA on the original surface.