Hamiltonian stability of Lagrangian tori in toric Kähler manifolds

Let (M,J,ω) be a compact toric Kähler manifold of dimℂ M=n and L a regular orbit of the Tn-action on M. In the present paper, we investigate Hamiltonian stability of L, which was introduced by Y.-G. Oh (Invent. Math.101, 501–519 (1990); Math. Z.212, 175–192) (1993)). As a result, we prove any regular orbit is Hamiltonian stable when (M,ω)=ℂℙn,ωFS) and (M,ω)=ℂℙn1× ℂℙn2,aωFS⊕ bωFS), where ωFS is the Fubini–Study Kähler form and a and b are positive constants. Moreover, they are locally Hamiltonian volume minimizing Lagrangian submanifolds.