Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

TL;DR

This paper extends Hamiltonian stability and mean curvature flow results for Lagrangian submanifolds to more general Kähler manifolds, including Fano manifolds, using weighted measures and variational methods.

Contribution

It introduces the concept of Hamiltonian $f$-stability for weighted volume functionals and demonstrates convergence of generalized Lagrangian mean curvature flow in Fano manifolds.

Findings

Hamiltonian $f$-stability characterized in toric Fano manifolds

Exponential convergence of flow to $f$-minimal Lagrangians

Extension of stability results to weighted measures in Kähler-Einstein manifolds

Abstract

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a K\"ahler-Einstein manifold to more general K\"ahler manifolds including a Fano manifold equipped with a K\"ahler form $\omega\in 2\pi c_1(M)$ by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold $L$ in a K\"ahler manifold $M$ and investigate the variational problem of $L$ for the weighted volume functional. We call a stationary point of the weighted volume functional $f$-minimal, and define the notion of Hamiltonian $f$-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an $f$-minimal and Hamiltonian $f$-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an $f$-minimal Lagrangian submanifold.