Generalized Lagrangian mean curvature flows: the cotangent bundle case

Knut Smoczyk Leibniz Universit├Ąt Hannover Mao-Pei Tsui The University of Toledo Mu-Tao Wang Columbia University

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1608.10027

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost K\"ahler 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.
mean curvature flows, cotangent bundle, Einstein connection
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  title={Generalized Lagrangian mean curvature flows: the cotangent bundle case},
  author={Knut Smoczyk, Mao-Pei Tsui, and Mu-Tao Wang},
Knut Smoczyk, Mao-Pei Tsui, and Mu-Tao Wang. Generalized Lagrangian mean curvature flows: the cotangent bundle case. 2016.
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