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The Gibbons-Hawking ansatz provides a large family of circle-invarianthyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, weprove versions of the Thomas conjecture on existence of special Lagrangianrepresentatives of Hamiltonian isotopy classes of Lagrangians, and theThomas-Yau conjecture on long-time existence of the Lagrangian mean curvatureflow. We also make observations concerning closed geodesics, curve shorteningflow and minimal surfaces.

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