Foliation structure for generalized Hénon mappings

We consider a generalized H\'{e}non mapping $f:\mathbb{C}^2\to\mathbb{C}^2$ and its Green function $g^+:\mathbb{C}^2\to\mathbb{R}_{\ge 0}$ (see Section 2). It is well known that each level set of the form $\{g^+=c\}$ 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 $\mathbb{P}^2$. Namely, for any holomorphic parametrization of any leaf, the Fubini-Study metric of $\mathbb{P}^2$ of its derivative is uniformly bounded. We also study the behavior of level sets of $g^+$ near infinity.