Lagrangian immersions in the product of Lorentzian two manifolds

Abstract

For Lorentzian two manifolds $(\Sigma _1,g_1)$ and $(\Sigma _2,g_2)$ we consider the two product para-Kähler structures $(G^{\epsilon },J,\Omega ^{\epsilon })$ defined on the product four manifold $\Sigma _1\times \Sigma _2$, with $\epsilon =\pm 1$. We show that the metric $G^{\epsilon }$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $\kappa _1,\kappa _2$ of $g_1,g_2$, respectively, are both constants satisfying $\kappa _1=-\epsilon \kappa _2$ (resp. $\kappa _1=\epsilon \kappa _2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product $\gamma _1\times \gamma _2\subset \Sigma _1\times \Sigma _2$, where $\gamma _1$ and $\gamma _2$ are geodesics. We study Lagrangian surfaces in the product $d{\mathbb S}^2\times d{\mathbb S}^2$ with parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and $H$-minimal surfaces.