On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Abstract

Let $(Σ_1,g_1)$ and $(Σ_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures $(G^ε,J,Ω^ε)$ on the product 4-manifold $Σ_1\timesΣ_2$ given by $ G^ε=g_1\oplus εg_2$, $ε=\pm 1$ and $J$ is the canonical product complex structure. Thus for $ε=1$ the Kähler metric $G^+$ is Riemannian while for $ε=-1$, $G^-$ is of neutral signature. We show that the metric $G^ε$ is locally conformally flat iff the Gauss curvatures $κ(g_1)$ and $κ(g_2)$ are both constants satisfying $κ(g_1)=-εκ(g_2)$. We also give conditions on the Gauss curvatures for which every $G^ε$-minimal Lagrangian surface is the product $γ_1\timesγ_2\subsetΣ_1\timesΣ_2$, where $γ_1$ and $γ_2$ are geodesics of $(Σ_1,g_1)$ and $(Σ_2,g_2)$, respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian $G^ε$-minimal surfaces.