The Wu metric and minimum ellipsoids (Q2752841)

The aim of the paper under this review is twofold. The author first gives a comprehensive and intuitive introduction to the Wu metric on Kobayashi hyperbolic complex manifold. The Wu metric was defined by \textit{H. Wu} [cf. Several Complex Variables, Math. Notes 38, 640-682 (1993; Zbl 0773.32017)] and it is defined on all complex manifolds that can be semi-definite. However, the Wu metric is definite on Kobayashi hyperbolic manifolds, and hence this case is specially important.NEWLINENEWLINENEWLINEIn this paper, the author gives fundamental properties of the Wu metric on Kobayashi hyperbolic manifolds such as continuity, invariance, distance-decreasing property and convex property. He gives concise proofs of several fundamental characteristic merits of the Wu metric, which is partly in conjunction of a conjecture of \textit{S. Kobayashi} [`Hyperbolic manifolds and holomorphic mappings', Marcel Dekker (1970; Zbl 0207.37902)]. Next the Wu metric on a Thullen domain is dealt with. The author discusses a curvature behavior of the Wu metric on Thullen domains following the joint work with Cheung. In addition, he investigates the asymptotic behavior of the Wu metric near the strongly pseudoconvex boundary point of a bounded domain of holomorphy after earlier work due to \textit{I. Graham} [Trans. Am. Math. Soc. 207, 219-240 (1975; Zbl 0305.32011)].NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].