Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space forms

TL;DR

This paper classifies Hamiltonian stationary Lagrangian surfaces in complex space forms, focusing on cases with constant or harmonic mean curvature, revealing conditions for parallel second fundamental form and providing a complete classification under certain curvature assumptions.

Contribution

It provides a complete classification of Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space forms under specific curvature conditions.

Findings

Second fundamental form is parallel when mean curvature is a non-zero constant.

Classification of surfaces with harmonic mean curvature and constant Gaussian curvature.

Identification of conditions for Hamiltonian stationary Lagrangian surfaces in complex space forms.

Abstract

In this paper, we study Hamiltonian stationary Lagrangian surfaces in complex space forms. We first show that when the mean curvature is a non-zero constant, the second fundamental form is parallel. We then consider the case in which the mean curvature is a non-constant harmonic function. Under the additional assumption that the Gaussian curvature is constant, we obtain a complete classification of such Lagrangian surfaces.