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-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 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.

11 pages. arXiv admin note: text overlap with arXiv:1305.1561

References in corpus (3)

• On the Geometry of Spaces of Oriented Geodesics
• On Indefinite Special Lagrangian Submanifolds in Indefinite Complex Euclidean Spaces
• On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Cited by in corpus (1)

• Almost Paracomplex Structures on 4-Manifolds