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HOME > Table of Contents and Abstracts > Vol. 67, No. 1
Tohoku Mathematical Journal
2015
March
SECOND SERIES VOL. 67, NO. 1
Tohoku Math. J.
67 (2015), 137-152
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Title
ON MINIMAL LAGRANGIAN SURFACES IN THE PRODUCT OF RIEMANNIAN TWO MANIFOLDS
Author
Nikos Georgiou
(Received May 20, 2013, revised December 2, 2013) |
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.
Mathematics Subject Classification.
Primary 53D12; Secondary 49Q05.
Key words and phrases.
Kähler structures, minimal submanifolds.
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