Local vs. global hyperconvexity, tautness or k-completeness for unbounded open sets in Cn

Abstract

Some known localization results for hyperconvexity, tautness or kcompleteness of bounded domains in C n are extended to unbounded open sets in C n .

Lipschitzness of the Lempert and Green functions

“…Moreover, recall (cf. [7], Proposition 3.2) that D is a taut domain if and only if lim z∈K,w→∂D l D (z, w) = 1 for any K ⋐ D.…”

“…Hence D is taut (cf. [7], page 607) and therefore hyperbolic. ¿From Corollary 6 we get that under the same assumptions gD and A D are Lipschitz functions (in both arguments).…”

“…On the other hand, if D is unbounded, then, by hyperbolicity, m * = lim inf z∈K,w→∞ l D (z, w) > 0 (use e.g. [7], Proposition 3.1). Fix a m ∈ (0, min{1/2, m * }).…”

Hyperbolicity in unbounded convex domains

“…In this direction, Gaussier [5] gave some conditions in terms of existence of peak and anti-peak functions at infinity for an unbounded domain to be hyperbolic, taut or complete hyperbolic. Recently, Nikolov and Pflug [14] deeply studied conditions at infinity which guarantee hyperbolicity, up to a characterization of hyperbolicity in terms of the asymptotic behavior of the Lempert function.…”

Remarks on Lempert functions of balanced domains

“…Since D is bounded, it is easily seen that for any neighborhood U of b there is another neighborhood V ⊂ U of b and a number s∈ (0, 1] such that, if ϕ ∈ O(D, D) with ϕ(0) ∈ V, then ϕ(sD) ⊂ D ∩ U.Now we choose U such that there is a negative plurisubharmonic function u on D ∩ U with lim D∋z→b u(z) = 0. Applying Proposition 3.4 in[3] implies that b is a t-point for D ∩ D which means that ( * ) (ϕ j | sD ) is compactly divergent w.r.t. D ∩ U.Assume now that (ϕ j ) does not converge to b.…”