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

TL;DR

The paper demonstrates that most Lagrangian tori arising as principal orbits in complex projective spaces are not volume minimizing under Hamiltonian isotopies when the complex dimension exceeds two, despite their minimality and stability.

Contribution

It proves that almost all Lagrangian torus orbits in ${ m CP}^n$ are not Hamiltonian volume minimizing for $n > 2$, revealing limitations of their volume-minimizing properties.

Findings

Most principal orbits are not volume minimizing under Hamiltonian isotopies.

These orbits are Hamiltonian minimal and stable.

The result holds for complex dimension greater than two.

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.