Complex geodesics in convex tube domains II

Abstract

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in C n containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the KobayashiRoyden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in C 2 and for those in C n whose logarithmic image is strictly convex in the geometric sense.

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

“…We find a fairly complete description of the continuous extension of complex geodesics up to the boundary in convex tube domains (see Theorem 12). We heavily rely on methods recently developed in [18] and [19].…”

“…Geodesics in convex tube domains with bounded bases. Below we use the notation from the papers [18] and [19] and we present some consequences of the results given there. We mainly concentrate on results for tube domains with bounded bases.…”

“…The theorem below follows directly from Theorem 3.3 in [18], Theorem 3.1 and Remark 3.3 in [19]. Let us draw the attention of the Reader to the fact that we restrict ourselves to the cases of tube domains with bounded bases.…”

“…But whether one may take as the last domain a convex one is not known (see [8]). With any convex domain D ⊂ R n we may relate the convex cone S(D) := {v ∈ R n : a + [0, ∞)v ⊂ D} with some (equivalently, any) a ∈ D (see e. g. [22], [19]).…”

“…Recall that in the proofs of positive results on Gromov hyperbolicity A. Zimmer extensively used the notions in (among others) the case of unbounded semitube domains (see [21]). And making use of recent results of Zajac ([18] and [19]) we present detailed general results on the boundary behavior of geodesics in tube domains over bounded smooth, strictly convex bases. In the above formula p denotes the Poincaré distance on the unit disc D, D := {λ ∈ C : |λ| < 1}.…”