Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

作者：Anciaux Henri; Guilfoyle Brendan; Romon Pascal^*

来源：Journal of Geometry and Physics, 2011, 61(1): 237-247.

DOI：10.1016/j.geomphys.2010.09.017

• Lagrangian surfaces
• Minimal surfaces
• Hamiltonian stationary surfaces
• Pseudo-Kahler metric

Given an oriented Riemannian surface (Sigma, g), its tangent bundle T Sigma enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure 2, a pseudo-metric G with neutral signature and a symplectic structure Omega. We give a local classification of those surfaces of T Sigma which are both Lagrangian with respect to Omega 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 TS2 or TH2 respectively. We relate the area of the congruence to a second-order functional F = f root H-2 - K dA on the original surface.

• 出版日期2011-1