The deformation of symplectic critical surfaces in a K\"ahler surface-I

In this paper we derive the Euler-Lagrange equation of the functional $L_\beta=\int_\Sigma\frac{1}{\cos^\beta\alpha}d\mu, ~~\beta\neq -1$ in the class of symplectic surfaces. It is $\cos^3\alpha {\bf{H}}=\beta(J(J\nabla\cos\alpha)^\top)^\bot$, which is an elliptic equation when $\beta\geq 0$. We call such a surface a $\beta$-symplectic critical surface. We first study the properties for each fixed $\beta$-symplectic critical surface and then prove that the set of $\beta$ where there is a stable $\beta$-symplectic critical surface is open. We believe it should be also closed. As a precise example, we study rotationally symmetric $\beta$-symplectic critical surfaces in ${\mathbb C}^2$ carefully .