Almost all standard Lagrangian tori in C^n are not Hamiltonian volume minimizing

TL;DR

This paper demonstrates that most standard Lagrangian tori in complex n-space are not volume minimizing under Hamiltonian isotopies when n > 2, and explores similar properties in toric Kähler manifolds.

Contribution

It proves that the majority of standard Lagrangian tori in C^n are not Hamiltonian volume minimizing for dimensions greater than two, advancing understanding of their geometric properties.

Findings

Most standard Lagrangian tori in C^n are not Hamiltonian volume minimizing for n > 2.

Discussion on existence of Hamiltonian non-volume minimizing Lagrangian tori in toric Kähler manifolds.

Provides insights into the geometric behavior of Lagrangian tori under Hamiltonian isotopies.

Abstract

In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangian torus orbits of compact toric Kahler manifolds.