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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2019, Issue 750

# Generalized Lagrangian mean curvature flows: The cotangent bundle case

Knut Smoczyk
• Institut für Differentialgeometrie and Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
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/ Mao-Pei Tsui
• Department of Mathematics, National Taiwan University, Taipei 106, Taiwan; National Center for Theoretical Sciences, Mathematics Division, 1 Sec. 4, Roosevelt Rd., Taipei 106, Taiwan; and Department of Mathematics and Statistics, University of Toledo, 2801 W. Bancroft St, Toledo, Ohio 43606-3390, USA
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/ Mu-Tao Wang
Published Online: 2016-08-30 | DOI: https://doi.org/10.1515/crelle-2016-0047

## Abstract

In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel $\left(n,0\right)$-form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

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Revised: 2016-07-18

Published Online: 2016-08-30

Published in Print: 2019-05-01

Funding Source: Simons Foundation

Award identifier / Grant number: 239677

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1105483

Award identifier / Grant number: DMS-1405152

The first named author was supported by the DFG (German Research Foundation). The second named author was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation #239677 and in part by Taiwan MOST grant 104-2115-M-002-001-MY2. This material is based upon work supported by the National Science Foundation under Grant Numbers DMS-1105483 and DMS-1405152 (Mu-Tao Wang).

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 750, Pages 97–121, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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