On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces

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 $n\le 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n\ge 3$. 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 $n\ge 3$ as well as the case of ${\mathbb{C}}^n$ and $\mathbb{C}{P}^n$.

26 pages, minor corrections