Talk Abstract: The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal submanifolds and mean curvature flow in these 4-manifolds, especially geodesics and minimal surfaces. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.