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

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 = 0, it is known that the sub-level sets of the Green's function G + H associated with H are Short C 2 's. For a given c > 0, we study the holomorphic automorphism group of such a Short C 2 , namely Ωc = {G + H < c}. The unbounded domain Ωc ⊂ C 2 is known to have smooth real analytic Levi-flat boundary. Despite the fact that Ωc admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, Aut(Ωc) 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 C 2 's to be biholomorphic.