Generalized Lagrangian mean curvature flows: the cotangent bundle case

TL;DR

This paper extends the theory of generalized Lagrangian mean curvature flows to cotangent bundles, proving long-term existence and stability results, and showing the canonical connection is Ricci flat and Einstein.

Contribution

It demonstrates that the canonical connection on cotangent bundles is Ricci flat, and establishes long-term existence and stability of the flow for zero sections.

Findings

Canonical connection on cotangent bundle is Ricci flat.

Flow preserves exactness and zero Maslov class.

Long-term existence and convergence to stable zero section.

Abstract

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.