On the hyperbolicity of projective plane with lacunary curves (Q1342806)

Let \(X\) be a complex manifold and \(M\) a dense subdomain of \(X\). Let \(d_M (p,q)\) denote the Kobayashi pseudodistance between points \(p, q \in M\). One extends \(d_M\) to \(X\) by defining \[ d_M (p,q) = \lim_{p' \to p, q' \to q} d_M(p', q') \qquad p', q' \in M. \] A point \(p \in X\) is called a degeneracy point of \(d_M\) on \(X\) is there exists a point \(q \in X \setminus \{p\}\) such that \(d_M(p,q) = 0\). The set of all degeneracy points is denoted by \(S_M (X)\). Here the author studies the structure of \(S_M (X)\) when \(X =\mathbb{C}^2\), \(A\) is a curve in \(X\), and \(M = X\setminus A\). In this case he proves Theorem: If \(S_M (X)\) is a curve in \(X\) then \(S_M(X)\) is composed of nonhyperbolic curves with respect to \(A\).