Convergence Of Mean Curvature Flow Research Articles

Convergence of mean curvature flow in hyper-Kähler manifolds

Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperk\"ahler 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 hyperk\"ahler complex structure. In particular, our result implies some kind of energy gap theorem for hyperk\"ahler manifolds which have no complex Lagrangian submanifolds.

The extension and convergence of mean curvature flow in higher codimension

In this paper, we investigate the convergence of the mean curvature flow of closed submanifolds in R n + q \mathbb {R}^{n+q} . We show that if the initial submanifold satisfies some suitable integral curvature conditions, then along the mean curvature flow it will shrink to a round point in finite time.