Advanced Search

Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler 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 ${cos}^3\alpha H={\left(J{(J\nabla cos\alpha )}^{\top }\right)}^{\perp }$, where $H$ is the mean curvature vector of $\Sigma$ 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...