Complete hyperbolic neighborhoods in almost-complex surfaces

Abstract

We prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is ${\bf R^4}$ with an arbitrary almost-complex structure $J$ of class $C^{1.\alpha}$. Let $C$ be a non-singular $J$-complex curve passing through the origin. Our result cah be stated as follows: There exists a basis $\{U_j\}$ of neighborhoods of zero in ${\bf R^4}$, such that $(U_j,J)$ are complete hyperbolic in the sence of Kobayashi, moreover $(U_j\setminus C,J)$ are complete hyperbolic as well. The fact that this result remains true for any almost-complex structure is somewhat suprising. Really, given any germ of a non-singular real surface $C\ni 0$ in ${\bf R^4}$, one can easily construct an almost-complex structure $J$ in a neighborhood of zero, such that $C$ becomes a $J$-complex curve. Typical corollary is the following: Let ${\cal M}_{\omega, 5l}$ be the Banach manifold consisting of pairs $(J,\{D_j\}_{j=1}^5)$, where $J$ is any almost-complex structure on ${\bf CP^2}$ tamed by the Fubini-Studi form $\omega$ and $\{D_j\}_{j=1}^5$ the union of five $J$-complex lines in ${\bf CP^2}$ in general position. The set ${\cal H}_{\omega, 5l}$ consisting of $(J, \{D_j\}_{j=1}^5)$ with $Y=({\bf CP^2}\setminus \bigcup_{j=1}^5 D_j,J)$ hyperbolically imbedded into $({\bf CP^2}, J)$ is an open nonempty subset of ${\cal M}{\omega, 5l}$.