Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

TL;DR

This paper classifies minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface, revealing explicit descriptions in flat cases and affine normal bundles over geodesics in non-flat cases, with applications to surface motions.

Contribution

It provides a local classification of minimal Lagrangian surfaces in the tangent bundle of Riemannian surfaces, including explicit descriptions and applications to surface motions in 3D space.

Findings

In non-flat cases, minimal Lagrangian surfaces are affine normal bundles over geodesics.

In flat cases, there exists a large explicit family of Lagrangian minimal surfaces.

Surface motions induce Hamiltonian motions of their normal congruences, linking geometry and dynamics.

Abstract

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $T\Sigma$ which are both Lagrangian with respect to $\Om$ 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 $T\S^2$ or $T \H^2$ respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.