An example of a strictly pseudoconvex domain homeomorphic to the ball with a complex disk attached from the outside (Q1387419)

Motivated by a question related to the Jacobian problem, the author establishes the following: There exist a complex manifold \(X\), an analytic branched covering \(f:X\to\mathbb{B}^4\) (where \(\mathbb{B}^4\) is the unit ball in \(\mathbb{C}^2)\), a strictly pseudoconvex domain \(\Omega\subset X\) homeomorphic to the four-dimensional ball, and a complex disk \(D\subset X\) such that (i) \(\Omega\) is contained in the regular part of the covering space \(X\); (ii) \(\partial D\subset\partial\Omega\), \(\text{int} D \subset X \setminus \overline \Omega\). That is, \(\Omega\) is a strictly pseudoconvex domain with a complex disk attached from the outside.