Lagrangian immersions in the product of Lorentzian two manifold

Abstract

For Lorentzian 2-manifolds $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ we consider the two product para-K\"ahler 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 curves of constant curvature. We study Lagrangian surfaces in the product $d{\mathbb S}^2\times d{\mathbb S}^2$ with non null parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and $H$-minimal surfaces.