Concave domains with trivial biholomorphic invariants

Abstract

Abstract. It is proved that if F is a convex closed set in C n , n ≥ 2, containing at most one (n − 1)-dimensional complex hyperplane, then the Kobayashi metric and the Lempert function of C n \ F identically vanish.Let D be a domain in These invariants can be characterized as the largest metric and function which decrease under holomorphic mappings and coincide with the Poincaré metric and distance on ∆. It is well known that if D is a bounded domain in C n , or a plane domain whose complement contains at least two points, then K D (z, X) > 0 for X = 0 and D (z, w) > 0 for z = w. On the other hand, the Kobayashi metric and the Lempert function of a plane domain whose complement contains at most one point identically vanish. Note also that there are domains in C n with bounded connected complements and non-vanishing Kobayashi metrics and Lempert functions. For example, if z 0 is a strictly pseudoconvex boundary point of a domain D in C n , n ≥ 2, then (cf.

Entire curves avoiding given sets in Cn

“…The first result is a generalization of the main part of Theorem 1 in [7]. More precisely, we prove the following result.…”

“…The equivalence of (i) and (iii) follows from the proof of Theorem 1 in [7]. For the convenience of the reader we repeat here the main idea of the proof of (iii) (i) is not satisfied we may assume (after a biholomorphic mapping) that F=A • C n-l, where the closed convex set A, properly contained in C, contains at least two points.…”

“…The equivalence of (i) and (iii) follows from the proof of Theorem 1 in [7]. For the convenience of the reader we repeat here the main idea of the proof of (iii) =⇒ (i).…”

“…Then we may assume that (cf. [2] and [7]) FcH, Ac{zEC2:Rezl>_l and Rez2_>0}, and, in addition, that Rec~j_>l, l<_j<_k.…”

“…Doing the induction step, we may assume that r Then, since F is convex and does not contain any complex line, after an affine change of coordinates one has that (cf. [2] and [7]) FcH:-{zEC2:Rezl< land Rez2_<-l}, co(aj) C{zEC 2 :Rezl -->0}, aj+i E {z E C 2 :Rez2_>0}.…”