TY - JOUR

T1 - Generalized Lagrangian mean curvature flows: The cotangent bundle case

AU - Smoczyk, Knut

AU - Tsui, Mao Pei

AU - Wang, Mu Tao

N1 - Funding information: 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).

PY - 2019

Y1 - 2019

N2 - 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 (n, 0)-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.

AB - 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 (n, 0)-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.

UR - http://www.scopus.com/inward/record.url?scp=85065064264&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1604.02936

DO - 10.48550/arXiv.1604.02936

M3 - Article

AN - SCOPUS:85065064264

VL - 2019

SP - 97

EP - 121

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 750

ER -