On the automorphism group of certain Short $\mathbb C^2$'s

TL;DR

This paper investigates the automorphism groups of certain unbounded domains in complex two-space derived from Hénon maps, revealing their structure, limitations, and conditions for biholomorphic equivalence.

Contribution

It characterizes the automorphism groups of Short ^2 domains associated with Hnon maps and provides criteria for their biholomorphic equivalence.

Findings

Automorphism groups are non-trivial but cannot be arbitrarily large.

Domains admit exhaustion by biholomorphic images of the unit ball.

Necessary and sufficient conditions for biholomorphic equivalence are established.

Abstract

For a H\'enon 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, 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.