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 ℂ𝐻^𝑛. We consider a standard Hamiltonian 𝑇^𝑛-action on ℂ𝐻^𝑛, and show that every Lagrangian 𝑇^𝑛-orbits in ℂ𝐻^𝑛 is H-stable when 𝑛 ≤ 2 and there exist infinitely many H-unstable 𝑇^𝑛-orbits when 𝑛 ≥ 3. On the other hand, we prove a monotone 𝑇^𝑛-orbit in ℂ𝐻^𝑛 is H-stable and rigid for any 𝑛. Moreover, we see almost all Lagrangian 𝑇^𝑛-orbits in ℂ𝐻^𝑛 are not Hamiltonian volume minimizing when 𝑛 ≥ 3 as well as the case of ℂ^𝑛 and ℂ𝑃^𝑛.