Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point
Let \(\Omega\) be a domain in an almost complex manifold \((M,J)\). A boundary point \(p\in\partial\Omega\) is called strongly \(J\)-pseudoconvex if there is a \(C^2\) local defining function whose Levi form is positive definite for the \(J\)-complex tangent vector space \(T_p(\partial\Omega)\cap J(T_p(\partial\Omega)\) at \(p\). In the paper the following problem is studied: to classify the domains \(\Omega\) in \((M^{2n},J)\) that admit a sequence \(\varphi_k\in\text{Aut}(\Omega,J)\) and a point \(q\in\Omega\) such that \(\varphi_k(q)\to p\), where \(p\in\partial\Omega\) is strongly \(J\)-pseudoconvex. The author solves the problem for \(n = 2, 3\). He proves that in the case \(n=2\) any such domain is isomorphic to the unit ball \(\mathbb B^2\subset\mathbb C^2\), while in the case \(n = 3\) it is isomorphic either to the unit ball \(\mathbb C^3\subset\mathbb C^3\) ot to one of a certain infinite family of domains in \(\mathbb R^6\) which is explicitely constructed in the paper. In the case \(n=2\) this result was proved, under a restriction, by Gaussier and Sukhov (preprint math. CV/0307335).
scientific article; zbMATH DE number 5140516
scientific article