Unbounded domains in \({\mathbb C}^2\) with non-compact automorphisms group (Q1412407)

Let \(D\) be a domain in \({\mathbb C}^2\) and let \(\Aut(D)\) be the holomorphic automorphism group. In Theorem 1.1 the authors prove that if \(P\) is a homogeneous polynomial of degree 2k with no harmonic terms, such that (0,0) is a convex boundary point of \(\Omega_P\), where \[ \Omega_P=\{(z,w)\in {\mathbb C}^2: \operatorname{Re} w +P(z,\overline{z})<0\}, \] then there exists a bounded domain \(D_P \simeq \Omega_P\) such that \(\partial D\) is smooth and real analytic except at most at one point \(p_0\) where the boundary is Lipschitz. Moreover \(\partial D\) is globally real analytic and smooth if and only if \(\Omega_P =\{(z,w)\in {\mathbb C}^2: \operatorname{Re} w +| z| ^{2k}<0\},\) and there exists a boundary curve through \(p_0\) consisting of limit points of orbits of internal points under \(\Aut(D)\). They also give a characterization of the unit ball in \({\mathbb C}^2\). As a consequence of Theorem 1.1 and Berteloot results they get the following theorem: if \(D\) is a \(C^{\infty}\)-smooth domain possessing a boundary orbit accumulation point \(p\) with a finite type pseudoconvex boundary neighborhood and convex \(h\)-model \(\Omega_P\). Then \(D\) is biholomorphic to a bounded domain with real analytic boundary everywhere but possibly at one point where the boundary is Lipschitz and it is \(C^1\)-smooth if and only if it is a pseudoellipsoid.