Let \(\Omega \subset \mathbb{C}^ n\), \(n \geq 2\), be a bounded convex domain and \(G\) the Lie group of biholomorphic automorphisms of \(\Omega\). What are the geometrical obstructions for a point \(p \in \partial \Omega\) to be in the topological closure of some \(G\)-orbit in \(\Omega\)? Levi- flatness near \(p\) together with entire smoothness of the boundary is an obstruction in the following sense: Assume that \(\partial \Omega\) is \(C^ \infty\)-smooth and Levi-flat in some neighborhood \(U\) of \(p\) in \(\mathbb{C}^ n\). If \(p \in \overline {Gq}\) for some \(q \in \Omega\), then \(\Omega\) is a product domain \(\mathbb{E} \times \Omega_ 1\), where \(\mathbb{E} \subset \mathbb{C}\) is the unit disc (Theorem 1). In particular \(p \notin \overline {Gq}\) if \(\partial \Omega\) is entirely \(C^ \infty\)-smooth. In the special situation \(n = 2\) the author introduces the notion of a boundary point of convex exponential type. It serves as a substitute for convexity of \(\Omega\) and locally Levi-flatness: If \(\Omega \subset \mathbb{C}^ 2\) is a bounded domain and \(\partial \Omega\) is \(C^ \infty\)- smooth in a neighborhood of a boundary point \(p\) of convex exponential type, then \(p \notin \overline {Gq}\) for \(q \in \Omega\) (Theorem 3).
The convex-scaling technique introduced by S. Frankel [Acta Math. 163, No. 1/2, 109-149 (1989; Zbl 0697.32016)] and modified by the author is the main tool for proving the results.