On the Kobayashi geometry of domains

Abstract

In this thesis we study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in ℂn. Specifically, we study the following subjects: Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains Ω1 ⊊ ℂn and Ω2 ⊊ ℂm, and points a ∊ Ω1 and b ∊ Ω2, find an explicit lower bound for dist(f(Ω1(r)), Ω2c) in terms of r, where Ω1(r) := {z ∊ Ω1 | dist(z, Ω1c) ≥ r} and f : Ω1 → Ω2 is any holomorphic map such that f(a) = b. This is motivated by the classical Schwarz lemma (i.e., Ω1 = Ω2 = ⅅ) which gives dist(f(sⅅ), ⅅc) ≥ 4-1(1-s)dist(a, ⅅc)dist(b, ⅅc). We extend this to the case where Ω1 and Ω2 are convex domains. In doing this, we make crucial use of the Kobayashi pseudodistance. Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in ℂn. This bears relation to Graham's well-known big-constant/small-constant bounds from above and below on convex domains. Graham's upper bounds are frequently not sharp. Our estimates improve these bounds. The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle. A weak notion of negative curvature for the Kobayashi distance on domains in ℂn: We introduce and study a property that we call visibility with respect to the Kobayashi distance, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property visibility domains. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains. Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends