On the Automorphism Group of Certain Short C2's

For a Henon map of the form H(x, y) = (y, p(y) - ax), where p is a polynomial of degree at least two and a not equal 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 Omega(c) = {G(H)(+) < c}. The unbounded domain Omega(c) subset of 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 C-2's to be biholomorphic.

Automated summary

This paper studies the sub-level sets of Henon maps and their associated holomorphic automorphism groups. It is found that although the sub-level sets have flat boundaries and can be exhausted by biholomorphic images of the unit ball, their automorphism groups are not too large.