Compactness theorems for sequences of pseudo-holomorphic coverings between domains in almost complex manifolds

Abstract

Our aim in this paper is to characterize smooth domains (D, J) and $$(D',J')$$ in almost complex manifolds of real dimension $$2n+2$$ with a covering orbit $$\{f_k (p)\}$$ , accumulating at a strongly pseudoconvex boundary point, for some $$(J,J')$$ -holomorphic coverings $$f_k : (D,J)\rightarrow (D', J')$$ and $$p\in D$$ . It was shown that such domains are both biholomorphic to a model domain, if the source domain (D, J) admits a bounded strongly J-plurisubharmonic exhaustion function. Furthermore, if the target domain $$(D',J')$$ is strongly pseudoconvex, then both (D, J) and $$(D',J')$$ are biholomorphic to the unit ball in $${\mathbb C}^{n+1}$$ with the standard complex structure. Our results can be considered as compactness theorems for sequences of pseudo-holomorphic coverings. Lin and Wong (Rocky Mt J Math 20(1):179–197, 1990) and Ourimi (Proc AMS 128(3):831–836, 2000) generalize for relatively compact domains in almost complex manifolds.