  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.
