Vol. 305, No. 2, 2020

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Convergence of mean curvature flow in hyper-Kähler manifolds

Keita Kunikawa and Ryosuke Takahashi

Vol. 305 (2020), No. 2, 667–691
DOI: 10.2140/pjm.2020.305.667

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.

mean curvature flow, hyper-Kähler manifolds, hyper-Lagrangian submanifolds
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 53C26
Received: 17 October 2018
Revised: 15 May 2019
Accepted: 15 November 2019
Published: 29 April 2020
Keita Kunikawa
Advanced Institute for Materials Research
Tohoku University
Ryosuke Takahashi
Research Institute for Mathematical Sciences
Kyoto University