Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

TL;DR

This paper proves the stability of the generalized Lagrangian mean curvature flow in cotangent bundles for graphs induced by closed 1-forms, relaxing previous conditions and establishing stability near special Lagrangian submanifolds.

Contribution

It introduces new derivative estimates that weaken initial conditions and remove the positive curvature requirement from prior work on the flow's stability.

Findings

Flow is stable near special Lagrangian submanifolds induced by closed 1-forms.

Weaker initial conditions than previous results.

Elimination of positive curvature condition in stability analysis.

Abstract

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in Smoczyk-Tsui-Wang's work. More precisely, we prove that if the graph induced by a closed $1$-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.