Dynamical Stability of Minimal Lagrangians in K\"ahler-Einstein Manifolds of Non-Positive Curvature

TL;DR

This paper proves that minimal Lagrangians in non-positively curved K"ahler-Einstein manifolds are dynamically stable under Lagrangian mean curvature flow, establishing a link between linear and nonlinear stability.

Contribution

It demonstrates the dynamical stability of minimal Lagrangians under mean curvature flow, extending linear stability results to nonlinear stability in non-positive curvature settings.

Findings

Flow starting near a minimal Lagrangian converges smoothly to it.

Stability holds for $C^1$-close, Hamiltonian isotopic Lagrangians.

Stability does not extend to $C^0$-closeness, as shown by Neves.

Abstract

It is known that minimal Lagrangians in K\"ahler--Einstein manifolds of non-positive scalar curvature are linearly stable under Hamiltonian deformations. We prove that they are also stable under the Lagrangian mean curvature flow, and therefore establish the equivalence between linear stability and dynamical stability. Specifically, if one starts the mean curvature flow with a Lagrangian which is $C^1$-close and Hamiltonian isotopic to a minimal Lagrangian, the flow exists smoothly for all time, and converges to that minimal Lagrangian. Due to the work of Neves [Ann. of Math. 2013], this cannot be true for $C^0$-closeness.