Let \(\Omega\) be a domain in \(\mathbb{C}^ 2\). The boundary \(b \Omega\) is said to be pseudo-convex and of finite type \((2m)\) at a point \(\zeta_ 0 \in b \Omega\) if there is a homogeneous subharmonic polynomial \(H(z)\) of degree \(2m\) without harmonic term such that in good holomorphic coordinates one has \(\zeta_ 0 = (0,0)\) and \(b \Omega = \{(w,z) | \text{Re} (w) + H(z) + 0 (| z |^{2m} + \text{Im} (w)) = 0\}\). Let \({\mathcal H} (\zeta_ 0)\) denote the set of such polynomials \(H\). Note that if \(H_ 1\), \(H_ 2 \in {\mathcal H} (\zeta_ 0)\) then \(H_ 1(z) = H_ 2 (\lambda z)\) for some \(\lambda \in \mathbb{C}^*\).
The author proves the following Theorem: Let \(\Omega\) be a domain in \(\mathbb{C}^ 2\) with boundary \(b \Omega\), pseudoconvex and of finite type at \(\zeta_ 0 \in b \Omega\). Assume there exists a sequence \(\{\varphi_ n\}\) of holomorphic automorphisms of \(\Omega\) and a point \(z_ 0\in \Omega\) such that \(\lim \varphi_ n (z_ 0) = \zeta_ 0\). Then \(\Omega\) is biholomorphically equivalent to \(\{(w,z) | \text{Re} (w) + H(z) < 0\}\) where \(H \in {\mathcal H} (\zeta_ 0)\).
The proof utilizes the scaling technique of S. Pinchuk [Proc. Symp. Pure Math. 52, Part 1, 151-161 (1991; Zbl 0744.32013)].