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.