Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space forms
Toru Sasahara
TL;DR
This work studies Hamiltonian stationary Lagrangian surfaces in complex space forms whose mean curvature is harmonic. It proves that when the mean curvature magnitude $|H|$ is a nonzero constant, the second fundamental form is parallel, and it provides an explicit classification for the nonconstant harmonic mean curvature case under the assumption of constant Gaussian curvature. The main result shows that such surfaces must satisfy $K=\epsilon=-1$ and are locally the Hopf projection of explicit Lagrangian immersions into the complex hyperbolic plane $\mathbb{C}H^2(-4)$, obtained via a model in $H_1^5(-1)\subset \mathbb{C}^3_1$ with a positive parameter $m$. This yields concrete geometric models and links between intrinsic curvature constraints and Hopf-fibration constructions in complex hyperbolic geometry.
Abstract
In this paper, we deal with Hamiltonian stationary Lagrangian surfaces in complex space forms whose mean curvature is a harmonic function. We prove that if the mean curvature is non-zero constant, then the second fundamental form is parallel. In the non-constant mean curvature case, under the condition of constant Gaussian curvature, we give an explicit description of all such surfaces.