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

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 $Ω_c = \{ G^+_H < c \}$. The unbounded domain $Ω_c \subset \mathbb C^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $Ω_c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, Aut$(Ω_c)$ 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.

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