The Second Variation of the Functional $L$ of Symplectic Critical Surfaces in Kähler Surfaces

Abstract

Let $M$ be a complete Kähler surface and $\Sigma $ be a symplectic surface which is smoothly immersed in $M$. Let $\alpha $ be the Kähler angle of $\Sigma $ in $M$. In the previous paper Han and Li (JEMS 12:505–527, 2010) 2010, we study the symplectic critical surfaces, which are critical points of the functional $L=\int _{\Sigma }\frac{1}{\cos \alpha }d\mu $ in the class of symplectic surfaces. In this paper, we calculate the second variation of the functional $L$ and derive some consequences. In particular, we show that, if the scalar curvature of $M$ is positive, $\Sigma $ is a stable symplectic critical surface with $\cos \alpha \ge \delta >0$, whose normal bundle admits a holomorphic section $X\in L^2(\Sigma )$, then $\Sigma $ is holomorphic. We construct symplectic critical surfaces in $\mathbf{C}^2$. We also prove a Liouville theorem for symplectic critical surfaces in $\mathbf{C}^2$.

Keywords

Symplectic critical surface / Holomorphic curve / Kähler surface