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

Abstract

In this paper we derive the Euler-Lagrange equation of the functional $L_{\beta }={\int }_{\Sigma }\frac{1}{{\cos }^{\beta }\alpha }d\mu ,\beta \ne -1$ in the class of symplectic surfaces. It is ${\cos }^3\alpha \mathbf{H}=\beta (J(J\nabla \cos \alpha {)}^{\top }{)}^{\perp }$, which is an elliptic equation when $\beta \ge 0$. 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 .

arXiv comment32 pages

Cited by in corpus (1)

• The deformation of symplectic critical surfaces in a Kähler surface-II---Compactness