Convergence of mean curvature flow in compact hyperkähler manifolds

Keita Kunikawa, Ryosuke Takahashi

Published 2018-08-21Version 1

Inspired by the work of Leung-Wan, we study the mean curvature flow in compact hyperk\"ahler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. Each hyper-Lagrangian submanifold admits a 2-sphere valued potential map, called the {\it complex phase}, which evolves according to the generalized harmonic map flow equation. We study the behavior of the mean curvature flow from the view point of the harmonic map flow theory, and show the long-time existence and exponential convergence to complex submanifolds when the initial Dirichlet energy of the complex phase is very small.