On minimal Lagrangian surfaces in the product of Riemannian two manifolds

TL;DR

This paper investigates minimal Lagrangian surfaces in the product of two Riemannian surfaces with different metric signatures, characterizing when the product manifold is conformally flat and analyzing the stability of certain minimal surfaces.

Contribution

It provides new conditions relating Gauss curvatures for conformal flatness and characterizes minimal Lagrangian surfaces as products of geodesics, also exploring their Hamiltonian stability.

Findings

Metric $G^{ ext{epsilon}}$ is conformally flat iff curvatures are constant and satisfy $ ext{curvature condition}$.

Minimal Lagrangian surfaces are products of geodesics under certain curvature conditions.

Analysis of Hamiltonian stability of minimal surfaces in the product manifold.

Abstract

Let $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical K\"ahler 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 K\"ahler 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.