The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. We compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behaviour is apriori known.
Let D ⊂ C n be a smoothly bounded pseudoconvex Levi corank one domain with defining function r, i.e., the Levi form ∂∂r of the boundary ∂D has at least (n − 2) positive eigenvalues everywhere on ∂D. The main goal of this article is to obtain bounds for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points p, q ∈ D in terms of parameters that reflect the Levi geometry of ∂D and the distance of these points to the boundary. Applications include an understanding of Fridman's invariant for the Kobayashi metric on Levi corank one domains, a description of the balls in the Kobayashi metric on such domains that are centered at points close to the boundary in terms of Euclidean data and the boundary behaviour of Kobayashi isometries from such domains.
This article considers isometries of the Kobayashi and Carathéod-ory metrics on domains in C n and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincaré's theorem about biholomorphic inequivalence of B n , the unit ball in C n and ∆ n , the unit polydisc in C n and then provide few examples which suggest that B n cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries f : D 1 → D 2 to the closures under purely local assumptions on the boundaries. As an application, we show that there is no isometry between a strongly pseudoconvex domain in C 2 and certain classes of weakly pseudoconvex finite type domains in C 2 .
We study the Wu metric for the non-convex domains of the formwhere 0 < m < 1/2. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs E2m are obtained. The Wu metric is then verified to be a continuous Hermitian metric on E2m which is real analytic everywhere except along the complex hypersurface Z = {(0, z2, . . . , zn) ∈ E2m}. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of m. This verifies a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45, 32H15.
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