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

Abstract

In this paper we consider the compactness of $\beta$-symplectic critical surfaces in a Kähler surface. Let $M$ be a compact Kähler 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 ${\Sigma }_{i^{\prime }}$ 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 $\mathcal{S}$.

arXiv comment23 pages

References in corpus (1)

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