Symplectic critical surfaces in Kähler surfaces

Details Symplectic critical surfaces in Kähler surfaces Symplectic critical surfaces in Kähler surfaces

2007-11-14

arxiv.org/abs/0711.2211v1

Mathematics

Differential Geometry

Scientific paper

Let $M$ be a K\"ahler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the K\"ahler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\int_{\Sigma}\frac{1}{\cos\alpha}d\mu$ in the class of symplectic surfaces. It is $\cos^3\alpha H=(J(J\nabla\cos\alpha)^\top)^\bot$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the K\"ahler form $\omega$ in $M$, which is an elliptic equation. We then study the properties of the equation.

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