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

Abstract

In this paper we consider the compactness of $\beta$-symplectic critical surfaces in a K\"ahler surface. Let $M$ be a compact K\"ahler surface and $\Sigma_i\subset M$ be a sequence of closed $\beta_i$-symplectic critical surfaces with $\beta_i\to\beta_0\in (0,\infty)$. Suppose the quantity $\int_{\Sigma_i}\frac{1}{\cos^q\alpha_i}d\mu_i$ (for some $q>4$) and the genus of $\Sigma_{i}$ are bounded, then there exists a finite set of points ${\mathcal S}\subset M$ and a subsequence $\Sigma_{i'}$ that converges uniformly in the $C^l$ topology (for any $l<\infty$) on compact subsets of $M\backslash {\mathcal S}$ to a $\beta_0$-symplectic critical surface $\Sigma\subset M$, each connected component of $\Sigma\setminus {\mathcal S}$ can be extended smoothly across ${\mathcal S}$.