TY - JOUR

T1 - CONVERGENCE OF MEAN CURVATURE FLOW IN HYPER-KÄHLER MANIFOLDS

AU - Kunikawa, Keita

AU - Takahashi, Ryosuke

N1 - Funding Information:
The authors wish to express their gratitude to Shigetoshi Bando for helpful conversations. We also would thank Ryoichi Kobayashi for pointing out Corollary 1.2. Kunikawa is supported by JSPS KAKENHI Grant Number JP19K14521, and Takahashi is supported by Grant-in-Aid for JSPS Fellows Number 16J01211.
Publisher Copyright:
© 2020, Mathematical Sciences Publishers.

PY - 2020/4

Y1 - 2020/4

N2 - Inspired by work of Leung and Wan (J. Geom. Anal. 17:2 (2007) 343–364), we study the mean curvature flow in hyper-Kahler 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-Kahler complex structure. In particular, our result implies some kind of energy gap theorem for hyper-Kahler manifolds which have no complex Lagrangian submanifolds.

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

KW - hyper-Kahler manifolds

KW - hyper-Lagrangian submanifolds

KW - mean curvature flow

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U2 - 10.2140/pjm.2020.305.667

DO - 10.2140/pjm.2020.305.667

M3 - Article

AN - SCOPUS:85097183448

SN - 0030-8730

VL - 305

SP - 667

EP - 691

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -