Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz (Q6561317)

It is conjectured that under mirror symmetry Lagrangians correspond to holomorphic connections on a complex bundle and special Lagrangians to Hermitian-Yang-Mills connections. Motivated by this and the Donaldson-Uhlenbeck-Yau theorem on Hermitian Yang-Mills connections, \textit{R. P. Thomas} [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 467--498 (2001; Zbl 1076.14525)] conjectured that a unique special Lagrangian exists in a given Hamiltonian isotopy class of Lagrangians if and only if a certain stability condition holds.\N\NSpecial Lagrangians are volume-minimizing, and it is thus natural to study them using the mean curvature flow. It is now well-known that the mean curvature flow preserves the Lagrangian condition. \textit{R. P. Thomas} and \textit{S. T. Yau} [Commun. Anal. Geom. 10, No. 5, 1075--1113 (2002; Zbl 1115.53054)] conjectured that certain stability conditions for a Hamiltonian isotopy class of a given Lagrangian should imply the long-time existence and convergence of Lagrangian mean curvature flow starting at the given Lagrangian.\N\NThe authors consider this problem on hyper-Kähler manifolds of dimension four given by the Gibbons-Hawking ansatz. This provides a large family of hyper-Kähler 4-manifolds. In this setting, the authors 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.