Generalized Hénon mappings and foliation by injective Brody curves

Abstract

We consider a finite composition of generalized Hénon mappings $\mathfrak{f}:{\mathbb{C}}^2\to {\mathbb{C}}^2$ and its Green function ${\mathfrak{g}}^{+}:{\mathbb{C}}^2\to {\mathbb{R}}_{\ge 0}$ (see Section 2). It is well known that each level set $\{{\mathfrak{g}}^{+}=c\}$ for $c>0$ is foliated by biholomorphic images of $\mathbb{C}$ and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in ${\mathbb{P}}^2$ (see Section 4). Namely, for any injective holomorphic parametrization of any leaf, its derivative is bounded over $\mathbb{C}$ with respect to the Fubini-Study metric of ${\mathbb{P}}^2$. We also study the behavior of the level sets of ${\mathfrak{g}}^{+}$ near infinity.