The constant curvature property of the Wu invariant metric

Abstract

We investigate the property of the Wu invariant metric on a certain class of psuedoconvex domains. We show that the Wu invariant Hermitian metric, which in general behaves as nicely as the Kobayashi metric under holomorphic mappings, enjoys the complex hyperbolic curvature property in such cases. Namely, the Wu invariant metric is Kähler and has constant negative holomorphic curvature in a neighborhood of the spherical boundary points for a large class of domains in C n .

On the upper semicontinuity of the Wu metric

“…The Wu pseudometric has been introduced by H. Wu in [Wu 1993] (and [Wu]). Various properties of the Wu metric may be found for instance in [Che-Kim 1996], [Che-Kim 1997], [Kim 1998], [Che-Kim 2003], [Juc 2002]. Nevertheless, it seems that even quite elementary properties of this metric are not completely understood, e.g.…”