Minimal Lagrangian Surfaces in the Tangent Bundle of a Riemannian Surface
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 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 3 1 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 = √ H2 − K dA on the original surface.
Guilfoyle, B. & Klingenberg, W., 2010. On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geometry and its Applications, 28(4), pp.454–468. Available at: http://dx.doi.org/10.1016/j.difgeo.2009.12.002.