Proper holomorphic self-mappings of Hartogs domains in \(\mathbb{C}^ 2\) (Q1310132)

The authors identify a class of bounded pseudoconvex Hartogs domains in \(\mathbb{C}^ 2\) for which any proper holomorphic self-map must be biholomorphic. The domains have the form \(\Omega=\{(z,w)\in\mathbb{C}^ 2\mid| w|^ 2+\varphi(z)<0\}\) with \(\varphi(z)\) a smooth function; the set where the Levi determinant vanishes can have no interior limit points in the intersection of \(\Omega\) with the \(z\) plane, and at weakly pseudoconvex boundary points either the Levi determinant vanishes to finite order or else \({\partial^ k\varphi\over\partial z^ k}(z_ 0)=0\) for all integers \(k>0\).