Symplectic critical surfaces in Kähler surfaces

Let $M$ be a Kähler surface and $\mathrm{\Sigma }$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler 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 Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of the symplectic critical surfaces.