Lagrangian F-stability of closed Lagrangian self-shrinkers

TL;DR

This paper investigates the stability properties of closed Lagrangian self-shrinkers in complex Euclidean space, revealing conditions under which they are F-stable or unstable based on topological and spectral criteria.

Contribution

It establishes new links between topological invariants, spectral properties, and F-stability of closed Lagrangian self-shrinkers, including the equivalence of F-stability types for certain cases.

Findings

Any closed Lagrangian self-shrinker with first Betti number > 1 is F-unstable.

Two-dimensional embedded closed Lagrangian self-shrinkers are F-unstable.

F-stability for Betti number one cases is characterized by spectral properties of the twisted Laplacian.

Abstract

In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is Lagrangian F-unstable. For a closed Lagrangian self-shrinker with first Betti number equal to one, we show that Lagrangian F-stability is equivalent to Hamiltonian F-stability. We also characterize Hamiltonian F-stability of a closed Lagrangian self-shrinker by its spectral property of the twisted Laplacian.