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.
We study the Wu metric on convex egg domains of the form E2m = z ∈ C n : |z1| 2m + |z2| 2 + . . . + |zn−1| 2 + |zn| 2 < 1 where m ≥ 1/2, m = 1. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be C 2 -smooth. Overall however, the Wu metric is shown to be continuous when m = 1/2 and even C 1 -smooth for each m > 1/2, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45. Key words and phrases. Wu metric, Kobayashi metric, negative holomorphic curvature. Both the authors were supported by the DST-INSPIRE Fellowship of the Government of India. 1 The term 'holomorphic curvature' stands precisely for the holomorphic sectional curvature and is said to be strongly negative, if it is bounded above by a negative constant.2 We shall reserve the term 'metric' for what was termed as a sub-metric in [13] and use the word 'distance'for what is termed as a 'metric' in general topology and metric geometry. 3 These are also referred to as 'complex ellipsoids' in the complex analysis literature. However, since the ellipsoid in Wu's construction is defined by a polynomial of degree two, we adopt a different terminology.
For a domain D ⊂ C n , n ≥ 2, let F k D (z) = KD(z)λ I k D (z) , where KD(z) is the Bergman kernel of D along the diagonal and λ I k D (z) is the Lebesgue measure of the Kobayashi indicatrix at the point z. This biholomorphic invariant was introduced by B locki and in this note, we study the boundary behaviour of F k D (z) near a finite type boundary point where the boundary is smooth, pseudoconvex with the corank of its Levi form being at most 1. We also compute its limiting behaviour near the boundary of certain other basic classes of domains.1991 Mathematics Subject Classification. 32F45, 32A07, 32A25.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.