Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

الأصل · EN

Given an oriented Riemannian surface (Σ, g), its tangent bundle TΣ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure, a pseudo-metric with neutral signature and a symplectic structure. We give a local classification of those surfaces of TΣ which are both Lagrangian with respect to and minimal with respect to. 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 ³ or ³₁ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in T§² or T ² respectively. We relate the area of the congruence to a second-order functional F=∫ √H²-K dA on the original surface.