Two intrinsic pseudo-metrics with pseudoconvex indicatrices and starlike circular domains (Q1085314)

The author constructs two biholomorphically invariant (infinitesimal) pseudometrics on the tangent bundle of an m-dimensional complex manifold M. These metrics agree with the Poincaré metric when M is the unit disk, and they have pseudoconvex indicatrices in each fiber. The first pseudometric, denoted \(B^ M\), is constructed from square-integrable holomorphic m-forms using techniques developed by the author and \textit{J. Burbea} [Kodai Math. J. 7, 133-152 (1984; Zbl 0562.32013]. For a symmetric bounded M domain in \({\mathbb{C}}^ m\), the author shows that \(B^ M\) is the same as the Carathéodory-Reiffen metric; thus \(B^ M\) does not always agree with the usual Bergman metric. The second pseudometric, denoted \(P^ M\), is constructed using negative plurisubharmonic functions. It decreases under holomorphic mapping, hence lies between the Carathéodory-Reiffen pseudometric and the Kobayashi-Royden pseudometric. Moreover a complete circular domain M in \({\mathbb{C}}^ m\) is pseudoconvex if and only if it coincides with the indicatrix of \(p^ M\) at 0 under the usual identification of \({\mathbb{C}}^ n\) with its tangent space at 0. The author gives some applications to holomorphic mappings between complete circular domains, he does not mention the relationships, if any, between \(p^ M\) and the intrinsic pseudodistance recently constructed by \textit{M. Klimek} [Bull. Soc. Math. Fr. 113, 231-240 (1985; Zbl 0584.32037)] also using negative plurisubharmonic functions.