Symplectic critical surfaces in K\"ahler surfaces

Let $M$ be a K"ahler surface and $\mathrm{\Sigma }$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the K"ahler angle of $\mathrm{\Sigma }$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L={\int }_{\mathrm{\Sigma }}\frac{1}{\cos \alpha }d\mu$ in the class of symplectic surfaces. It is ${\cos }^3\alpha H=(J(J\mathrm{\nabla }\cos \alpha {)}^{\mathrm{\top }}{)}^{\mathrm{\perp }}$, where $H$ is the mean curvature vector of $\mathrm{\Sigma }$ 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.