On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces

TL;DR

This paper studies the Hamiltonian stability of Lagrangian tori in complex hyperbolic spaces, revealing stability properties depend on the dimension and orbit type, with many not being volume minimizing in higher dimensions.

Contribution

It characterizes Hamiltonian stability of Lagrangian T^n-orbits in complex hyperbolic spaces, highlighting dimension-dependent stability and rigidity results.

Findings

Lagrangian T^n-orbits are H-stable for n ≤ 2

Existence of infinitely many H-unstable T^n-orbits for n ≥ 3

Almost all T^n-orbits are not Hamiltonian volume minimizing when n ≥ 3

Abstract

In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n\leq 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n\geq 3$. On the other hand, we prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any $n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are not Hamiltonian volume minimizing when $n\geq 3$ as well as the case of $\mathbb{C}^n$ and $\mathbb{C}P^n$.