The deformation of symplectic critical surfaces in a Kähler surface-I

Abstract

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