Convergence of mean curvature flow in compact hyperk\"ahler manifolds

Inspired by the work of Leung-Wan, we study the mean curvature flow in compact hyperk$\"a$hler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the $\it\text{twistor energy}$ by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperk$\"a$hler complex structure. In particular, our result implies some kind of energy gap theorem for hyperk$\"a$hler manifolds which have no complex Lagrangian submanifolds.