Almost all Lagrangian torus orbits in $${\mathbb C}P^n$$ C P n are not Hamiltonian volume minimizing

Abstract

All principal orbits of the standard Hamiltonian \(T^n\)-action on the complex projective space \({\mathbb C}P^n\) are Lagrangian tori. In this article, we prove that most of them are not volume minimizing under Hamiltonian isotopies of \({\mathbb C}P^n\) if the complex dimension n is greater than two, although they are Hamiltonian minimal and Hamiltonian stable.