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On the automorphism group of certain Short $\mathbb C^2$'s (2108.07475v2)

Published 17 Aug 2021 in math.CV

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 $\mathbb C2$'s. For a given $c > 0$, we study the holomorphic automorphism group of such a Short $\mathbb C2$, namely $\Omega_c = { G+_H < c }$. The unbounded domain $\Omega_c \subset \mathbb C2$ 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 $\mathbb C2$'s to be biholomorphic.

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