Hamiltonian F-stability of complete Lagrangian self-shrinkers

TL;DR

This paper investigates the stability properties of Lagrangian self-shrinkers under Hamiltonian F-stability, providing a characterization theorem based on spectral analysis of the drifted Laplacian for certain complete submanifolds.

Contribution

It establishes a new characterization of Hamiltonian F-stability for complete Lagrangian self-shrinkers using eigenvalues and eigenspaces of the drifted Laplacian, under specific geometric conditions.

Findings

Characterization of Hamiltonian F-stability via eigenvalues of the drifted Laplacian

Conditions on the second fundamental form for stability analysis

Extension of stability criteria to complete Lagrangian self-shrinkers

Abstract

In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of $n$-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form satisfying the condition that there exist constants $C_0>0$ and $\varepsilon<\frac{1}{16n}$ such that $|A|^2\leq C_0+\varepsilon |x|^2$. We characterize the Hamiltonian F-stablity by the eigenvalues and eigenspaces of the drifted Laplacian.