On the Automorphism Group of Certain Short ℂ2’s

Abstract

Abstract For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not = 0$, it is known that the sub-level sets of the Green’s function $G^+_H$ associated with $H$ are Short $\mathbb {C}^2$’s. For a given $c> 0$, we study the holomorphic automorphism group of such a Short $\mathbb {C}^2$, namely $\Omega _c = \{ G^+_H < c \}$. The unbounded domain $\Omega _c \subset \mathbb {C}^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $\Omega _c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, $\textrm {Aut}(\Omega _c)$, cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short $\mathbb {C}^2$’s to be biholomorphic.