Embedding strictly pseudoconvex domains into balls

Source abstract

Every relatively compact strictly pseudoconvex domain D D with C 2 {{\mathbf {C}}^2} boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n ≥ 2 n \geq 2 there exist bounded strictly pseudoconvex domains D D in C n {\mathbb {C}^n} with real-analytic boundary such that no proper holomorphic map from D D into any finite dimensional ball extends smoothly to D ¯ \overline D .