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

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...

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Bibliographic Details
Main Authors Lee, Ping-Hung, Tsai, Chung-Jun
Format Journal Article
LanguageEnglish
Published 06.06.2024
Subjects
Mathematics - Differential Geometry
Mathematics - Symplectic Geometry
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Summary: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.
DOI:10.48550/arxiv.2406.04602