Inspired by work of Leung and Wan (J. Geom. Anal. 17:2 (2007) 343–364), we study the mean curvature flow in hyper-Kahler 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 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 hyper-Kahler complex structure. In particular, our result implies some kind of energy gap theorem for hyper-Kahler manifolds which have no complex Lagrangian submanifolds.
- hyper-Kahler manifolds
- hyper-Lagrangian submanifolds
- mean curvature flow