Kobayashi Distance Research Articles

INTERSECTIONS OF HOLOMORPHIC RETRACTS IN BANACH SPACES

AbstractUsing the Kobayashi distance, we provide sufficient conditions for the intersection of a family of holomorphic retracts in a Banach space to also be a holomorphic retract.

Noncommutative hyperbolic geometry on the unit ball of [formula omitted

In this paper we introduce a hyperbolic (Poincaré–Bergman type) distance δ on the noncommutative open ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 < 1 } , where B ( H ) is the algebra of all bounded linear operators on a Hilbert space H . It is proved that δ is invariant under the action of the free holomorphic automorphism group of [ B ( H ) n ] 1 , i.e., δ ( Ψ ( X ) , Ψ ( Y ) ) = δ ( X , Y ) , X , Y ∈ [ B ( H ) n ] 1 , for all Ψ ∈ Aut ( [ B ( H ) n ] 1 ) . Moreover, we show that the δ-topology and the usual operator norm topology coincide on [ B ( H ) n ] 1 . While the open ball [ B ( H ) n ] 1 is not a complete metric space with respect to the operator norm topology, we prove that [ B ( H ) n ] 1 is a complete metric space with respect to the hyperbolic metric δ. We obtain an explicit formula for δ in terms of the reconstruction operator R X : = X 1 ∗ ⊗ R 1 + ⋯ + X n ∗ ⊗ R n , X : = ( X 1 , … , X n ) ∈ [ B ( H ) n ] 1 , associated with the right creation operators R 1 , … , R n on the full Fock space with n generators. In the particular case when H = C , we show that the hyperbolic distance δ coincides with the Poincaré–Bergman distance on the open unit ball B n : = { z = ( z 1 , … , z n ) ∈ C n : ‖ z ‖ 2 < 1 } . We obtain a Schwarz–Pick lemma for free holomorphic functions on [ B ( H ) n ] 1 with respect to the hyperbolic metric, i.e., if F : = ( F 1 , … , F m ) is a contractive ( ‖ F ‖ ∞ ⩽ 1 ) free holomorphic function, then δ ( F ( X ) , F ( Y ) ) ⩽ δ ( X , Y ) , X , Y ∈ [ B ( H ) n ] 1 . As consequences, we show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ on [ B ( H ) n ] 1 . The results of this paper are presented in the more general context of Harnack parts of the closed ball [ B ( H ) n ] 1 − , which are noncommutative analogues of the Gleason parts of the Gelfand spectrum of a function algebra.

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.

THE SYMMETRIZED BIDISC AND LEMPERT'S THEOREM

Let G ⊆ C2 be the open symmetrized bidisc, namely G = {(λ1 + λ2, λ1λ2) : |λ1| < 1, |λ2| < 1}. In this paper, a proof is given that G is not biholomorphic to any convex domain in C2. By combining this result with earlier work of Agler and Young, the author shows that G is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set. 2000 Mathematics Subject Classification 32F45 (primary), 15A18 (secondary).

A GENERAL WOLFF THEOREM FOR ARBITRARY BANACH SPACES

The Kobayashi distance is used to generalise the classical theorem of Wolff to compact holomorphic fixed-point-free mappings on the open unit ball of an arbitrary complex Banach space E and more generally on bounded convex domains in E, thereby extending results of Abate for C n . This is compared to earlier results on bounded symmetric domains. The boundary behaviour of the Kobayashi distance one or both of z, w tend to the boundary.

Complex geometry of generalized annuli

We study the complex geometry of a class of domains in C n which generalize the annuli in C, i.e., which are quotients of the unit ball B n of C n for the action of a group gen- erated by a hyperbolic element of AutB n . In particular, we prove that the degree of holomor- phic maps between two such domains is bounded by a constant which depends on the ''radii'' of the domains only and we give some results on the existence of complex geodesics for the Kobayashi distance in these domains.

Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group

We present a characterization of the open unit ball in a separable infinite dimensional Hilbert space by the property of automorphism orbits among the domains that are not necessarily bounded. This generalizes the recent work of Kim and Krantz [6]. Key new features of this article include: a lower bound estimate of the Kobayashi metric and distance near a pluri-subharmonic peak boundary point of the domains in Banach spaces, an effective localization argument, and an improvement of weak-type convergence of sequences of biholomorphic mappings of domains in Banach spaces.

The Lindelöf principle and angular derivatives in convex domains of finite type

AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

Total sets and semicontinuity of the Kobayashi distance

Total sets and semicontinuity of the Kobayashi distance

A Schwarz Lemma for the Symmetrized Bidisc

Let φ be an analytic function from [ ] to the symmetrized bidisc (formula here) We show that if φ(0) = (0,0) and φ(λ) = ( s , p ) in the interior of Γ, then (formula here) Moreover, the inequality is sharp: we give an explicit formula for a suitable φ in the event that the inequality holds with equality. We show further that the inverse hyperbolic tangent of the left-hand side of the inequality is equal to both the Caratheodory distance and the Kobayashi distance from (0,0) to ( s , p ) in int Γ

The weak lower semicontinuity of the Kobayashi distance and its applications

In this paper we prove the lower semicontinuity with respect to the weak topology of the Kobayashi distance in a bounded, convex and open subset of a reflexive Banach space. We apply this result to the Denjoy-Wolff theorem for condensing mappings in the unit open ball in a strictly convex reflexive Banach space with the Kadec-Klee property.

A few properties of the Kobayashi distance and their applications

Let $N$ be a norming set in a Banach space $X$. In this paper we prove the lower semicontinuity with respect to the topology $\sigma( X,N)$ of the Kobayashi distance in a bounded, relatively compact in $\sigma ( X,N) $, convex and open subset of a Banach space. We apply this result to the Denjoy-Wolff type theorem.

Domains which are locally uniformly linearly convex in the Kobayashi distance

We show a construction of domains in complex reflexive Banach spaces which are locally uniformly convex in linear sense in their Kobayashi distance. We also show connections between norm and Kobayashi distance properties.

Local uniform linear convexity with respect to the Kobayashi distance

We introduce the notion of local uniform linear convexity of bounded convex domains with respect to their Kobayashi distances.

Transfer of Estimates from Convex to Strongly Pseudoconvex Domains in $ℂ^N$

In this article, estimates of the hyperbolic and Carathéodory distances in domains $G ⊂ ⊂ ℂ^n$, n ≥ 1, are obtained. They are equally valid for the Kobayashi distance.

Localization of the Kobayashi Distance

Localization of the Kobayashi Distance

Intrinsic metrics in complete circular domains

The purpose of this paper is to study the behavior of the Kobayashi and Carath6odory pseudodistances and (infinitesimal) pseudometrics near the origin of complete circular domain in a complex locally convex separate topological vector space. Let V be a locally convex separate topological vector space over the field C of the complex numbers. We recall that a domain D c V (that is, a connected open subset of V) is complete circular if CzeD whenever z e D and C~C, ICI < 1. For any domain D c V we denote by k o and c o respectively the Kobayashi and Carath6odory distance functions, and by x D and ?o the corresponding (infinitesimal) metrics. For the definitions and the principal properties of these objects see [4]. Our first result is the following:

Holomorphic Maps Which Preserve Intrinsic Metrics or Measures

Suppose that $M$ is a domain in a taut complex manifold $M’$, and that $\Omega$ is a strictly convex bounded domain in ${{\mathbf {C}}^n}$. We consider the following question: given a holomorphic map $F:M \to \Omega$ which is an isometry for the infinitesimal Kobayashi metric at one point, must $F$ be biholomorphic? With an additional technical assumption on the behavior of the Kobayashi distance near points of $\partial M$, we show that $F$ gives a biholomorphism of $M$ with an open dense subset of $\Omega$. Moreover, $F$ extends as a homeomorphism from a larger domain $\tilde M$ to $\Omega$. We also give some related results—refinements of theorems of Bland and Graham and Fornaess and Sibony, and the answer to a question of Graham and Wu.

Holomorphic maps which preserve intrinsic metrics or measures

Suppose that M M is a domain in a taut complex manifold M ′ M’ , and that Ω \Omega is a strictly convex bounded domain in C n {{\mathbf {C}}^n} . We consider the following question: given a holomorphic map F : M → Ω F:M \to \Omega which is an isometry for the infinitesimal Kobayashi metric at one point, must F F be biholomorphic? With an additional technical assumption on the behavior of the Kobayashi distance near points of ∂ M \partial M , we show that F F gives a biholomorphism of M M with an open dense subset of Ω \Omega . Moreover, F F extends as a homeomorphism from a larger domain M ~ \tilde M to Ω \Omega . We also give some related results—refinements of theorems of Bland and Graham and Fornaess and Sibony, and the answer to a question of Graham and Wu.

Pseudodistances and Pseudometrics on real and complex manifolds

In this paper we study the relationships between a class of distances and infinitesimal metrics on real and complex manifolds and their behavior under differentiable and holomorphic mappings. Some application to Riemannian and Finsler geometry are given and also new proofs and generalizations of some results of Royden, Harris and Reiffen on Kobayashi and Caratheodory metrics on complex manifolds are obtained. In particular we prove that on every complex manifold (finite or infinite- dimensional) the Kobayashi distance is the integrated form of the corresponding infinitesimal metric.