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Tohoku Mathematical Journal
2015
March
SECOND SERIES VOL. 67, NO. 1
Tohoku Math. J.
67 (2015), 137152

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 2dimensional Riemannian manifolds. Consider the two canonical Kähler structures \linebreak $(G^{\epsilon},J,\Omega^{\epsilon})$ on the product 4manifold $\Sigma_1\times\Sigma_2$ 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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