Minimal and Hamiltonian-minimal submanifolds in toric geometry

TL;DR

This paper explores Hamiltonian-minimal Lagrangian submanifolds in complex and toric manifolds, providing new conceptual proofs of their minimality and extending results to moment-angle manifolds.

Contribution

It introduces a new conceptual approach to proving H-minimality and minimality of certain submanifolds in symplectic toric geometry.

Findings

Proves H-minimality of submanifolds in complex and toric manifolds.

Establishes minimality of these submanifolds in moment-angle manifolds.

Provides a conceptual framework for understanding Hamiltonian-minimality.

Abstract

In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real quadrics. In particular, we explain the nature of this phenomenon by proving H-minimality in a more conceptual way, and prove minimality of the same submanifolds in the corresponding moment-angle manifolds.