Vol. 305, No. 2, 2020

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
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
Abstract

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.

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