Symplectic critical surfaces in Kähler surfaces

Abstract

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 ${\cos }^3\alpha H=(J(J\nabla \cos \alpha {)}^{\top }{)}^{\perp }$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, and $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$; it is an elliptic equation. We call a surface satisfying a this equation a symplectic critical surface. We show that, if $M$ is a Kähler–Einstein surface with a nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of symplectic critical surfaces. By our formula and Webster’s formula, we deduce that the Kähler angle of a compact symplectic critical surface is constant, which is not true a for noncompact symplectic critical surfaces.