Tohoku Mathematical Journal

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Nikos Georgiou

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Abstract

Let $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ be connected, complete and orientable 2-dimensional Riemannian manifolds. Consider the two canonical Kähler structures \linebreak $(G^{\epsilon},J,\Omega^{\epsilon})$ 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 if and only if 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.

Article information

Source
Tohoku Math. J. (2), Volume 67, Number 1 (2015), 137-152.

Dates
First available in Project Euclid: 20 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1429549583

Digital Object Identifier
doi:10.2748/tmj/1429549583

Mathematical Reviews number (MathSciNet)
MR3337967

Zentralblatt MATH identifier
1331.53112

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12]

Keywords
Kähler structures minimal submanifolds

Citation

Georgiou, Nikos. On minimal Lagrangian surfaces in the product of Riemannian two manifolds. Tohoku Math. J. (2) 67 (2015), no. 1, 137--152. doi:10.2748/tmj/1429549583. https://projecteuclid.org/euclid.tmj/1429549583


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