Generalized Lagrangian mean curvature flows: The cotangent bundle case

authored by

Knut Smoczyk, Mao Pei Tsui, Mu Tao Wang

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 (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.

Organisation(s)

Institut für Differentialgeometrie
Riemann Center for Geometry and Physics

External Organisation(s)

National Taiwan University
University of Toledo
Columbia University
National Center for Theoretical Sciences, Physics (NCTS)

Type

Artikel

Journal

Journal fur die Reine und Angewandte Mathematik

Volume

2019

Pages

97-121

No. of pages

25

ISSN

0075-4102

Publication date

2019

Publication status

Veröffentlicht

Peer reviewed

Yes

ASJC Scopus Sachgebiete

Mathematik (insg.), Angewandte Mathematik

Electronic version(s)

https://doi.org/10.48550/arXiv.1604.02936 (Access: Offen)
https://doi.org/10.1515/crelle-2016-0047 (Access: Geschlossen)