A domain in \(\mathbb{C}^ m\) not containing any proper image of the unit disc (Q2365039)

An example is given of a bounded domain in \(\mathbb{C}^m\), \(m\geq 2\), which contains no proper analytic discs. It is derived from a greater pathology, of a topological nature, which is observed in this domain. Previously it was known that \(B^n\) (the unit ball of \(\mathbb{C}^n\)), \(n\geq 1\), can be mapped properly and holomorphically into an arbitrary smooth bounded domain in one co-dimension. This example shows that a simultaneous assumption is necessary in any given co-dimension. In all known constructions of proper holomorphic maps from \(B^n\), \(n\geq 1\), the map is pushed toward the boundary of the target domain in a neighborhood which is assumed to have some degree of pseudoconvexity. The example presented in this paper indicates that such an assumption is probably necessary.