Convergence of mean curvature flow in hyperk\"ahler manifolds

TL;DR

This paper proves that mean curvature flow starting from hyper-Lagrangian submanifolds with small twistor energy in hyperk"ahler manifolds exists globally and converges to complex Lagrangian submanifolds, revealing an energy gap phenomenon.

Contribution

It introduces the concept of twistor energy and demonstrates convergence of mean curvature flow under small twistor energy in hyperk"ahler manifolds.

Findings

Flow exists for all time under small twistor energy

Flow converges to complex Lagrangian submanifolds

Energy gap theorem for hyperk"ahler manifolds without complex Lagrangians

Abstract

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.