Abstract: Lagrangian mean curvature flow is a nonlinear parabolic PDE with links to symplectic topology, Riemannian and complex geometry, and theoretical physics. A key open problem in the field is the Thomas-Yau conjecture, which gives a putative criterion for long-time existence and convergence of the flow. I will discuss a recent proof of a version of the Thomas-Yau conjecture for a family of 4-dimensional manifolds, which are gravitational instantons given by the Gibbons-Hawking ansatz. This is joint work with Goncalo Oliveira.