On the Compactification of Concave Ends

The concave end of a 1-corona ρ : X →]a, b[ always can be compactified if n := dimX ≥ 3. This was proved by Rossi [Ro] and Andreotti-Siu [AS]. For n = 2 this not true in general, as shown by a counterexample of Grauert, Andreotti-Siu and Rossi [AS, Gr, Ro]. However, if the concave end of a 1-corona ρ : X →]a, b[ is even hyperconcave (i.e. a = −∞), then this is again true also for dimX = 2. This was proved by Marinescu-Dinh [MD]. The Andreotti-Vesentini separation theorem [AV] implies the following necessary condition: If the concave end of a 1-corona ρ : X →]a, b[ can be compactified, dimX ≥ 2, then H(X) is Hausdorff, i.e. the space of exact C 0,1-forms is closed with respect to uniform convergence on compact sets together with all derivatives. In the present paper we show that this condition is also sufficient. We prove: