Hamiltonian minimal Lagrangian submanifolds in toric varieties

TL;DR

This paper extends the construction of Hamiltonian minimal Lagrangian submanifolds from complex space to toric varieties, contributing to symplectic geometry and the study of minimal submanifolds.

Contribution

It introduces a new method for constructing H-minimal Lagrangian submanifolds within toric varieties, expanding previous work from complex space.

Findings

Defined H-minimal submanifolds in toric varieties.

Extended previous constructions from complex space.

Provided new examples of minimal Lagrangian submanifolds.

Abstract

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebrated Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism. In the previous works the authors defined and studied a family of H-minimal Lagrangian submanifolds in complex space arising from intersections of Hermitian quadrics. Here we extend this construction to define H-minimal submanifolds in toric varieties.