Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.