On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Abstract

Let $({\Sigma }_1,g_1)$ and $({\Sigma }_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures $(G^{\epsilon },J,{\Omega }^{\epsilon })$ on the product 4-manifold ${\Sigma }_1\times {\Sigma }_2$ given by $G^{\epsilon }=g_1\oplus \epsilon g_2$, $\epsilon =\pm 1$ and $J$ is the canonical product complex structure. Thus for $\epsilon =1$ the Kähler metric $G^{+}$ is Riemannian while for $\epsilon =-1$, $G^{-}$ is of neutral signature. We show that the metric $G^{\epsilon }$ is locally conformally flat iff the Gauss curvatures $\kappa (g_1)$ and $\kappa (g_2)$ are both constants satisfying $\kappa (g_1)=-\epsilon \kappa (g_2)$. We also give conditions on the Gauss curvatures for which every $G^{\epsilon }$-minimal Lagrangian surface is the product ${\gamma }_1\times {\gamma }_2\subset {\Sigma }_1\times {\Sigma }_2$, where ${\gamma }_1$ and ${\gamma }_2$ are geodesics of $({\Sigma }_1,g_1)$ and $({\Sigma }_2,g_2)$, respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian $G^{\epsilon }$-minimal surfaces.

23 pages

References in corpus (3)

• On the Geometry of Spaces of Oriented Geodesics
• Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface
• A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

Cited by in corpus (4)

• A new geometric structure on tangent bundles
• Almost Paracomplex Structures on 4-Manifolds
• Lagrangian immersions in the product of Lorentzian two manifold
• A para-Kaehler structure in the space of oriented geodesics in a real space form