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