Short $\mathbb{C}^k$
John Erik Fornæss · Advanced studies in pure mathematics · 2019
One of Oka's main contributions was to solve the Levi problem.There are various ways to generalize the Levi Problem.The Union problem is one:Suppose that each ni is Stein.Is n Stein?To approach the Union Problem, one can try at first to understand the simplest cases of n.Example 1.1.Long C 2 • Suppose that each Oi is biholomorphic to C 2 .Then we call 0 a long C 2 • It is an open question whether all long C 2 are actually biholomorphic to C 2 0 Example 1.2.(Fornress, ([F, 1976])) In dimension 3 and higher it can happen that 0 is not Stein and that each On is biholomorphic to a ball.This left open the question in dimension 2.Theorem 1.3.(Fornress-Sibony, ([FS, 1981])) Suppose that each Oi is biholomorphic to the unit ball in C 2 • If the (infinitesimal} Kobayashi metric of n is not identically zero, then n is biholomorphic to the ball or to~ x C, where~ is the unit disc.Recall that the (infinitesimal) Kobayashi metric of n vanishes identically if and only if for all p E 0 and any tangent vector ,; to 0 at p and for any R > 0, there exists a holomorphic map f :This theorem left still open the case when the Kobayashi metric vanishes identically.The most obvious example of such a case is when n = C 2 • However, the question remaining was whether there was any