Kobayashi Distance Research Articles

Comparison of the real and the complex Green functions, and sharp estimates of the Kobayashi distance

We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also give some lower estimates. In order to obtain those results, and because it is of independent interest, we refine and unify some classical estimates about the Kobayashi distance and the Lempert function in such domains.

On the Gromov Hyperbolicity of Convex Domains in $${\mathbb {C}}^n$$ C n

We prove that if a \({\mathcal {C}}^\infty \)-smooth bounded convex domain in \({\mathbb {C}}^n\) contains a holomorphic disc in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also give examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs Bn(H) in B(H)n1+⋯+nk, where n=(n1,…,nk)∈Nk and B(H) is the algebra of all bounded linear operators on a Hilbert space H. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on Bn(H), study their properties, and obtain hyperbolic versions of Schwarz–Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case n1=⋯=nk=1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk) of all free holomorphic automorphisms of Dk(H), and the δH-topology induced on Dk(H) is the usual operator norm topology. The restriction of δH to the scalar polydisk Dk is equivalent to the Kobayashi distance on Dk. Most of the results of this paper are presented in the more general setting of Harnack (resp. Poisson) parts of the closed polyball Bn(H)−.

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

Abstract We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains. The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in ℝ n {\mathbb{R}^{n}} with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.

On the Gromov non-hyperbolicity of certain domains in [formula omitted

In this paper, we prove that if Ω is a bounded convex domain in Cn, n≥2, and S is an affine complex hyperplane such that Ω∩S is not empty, then Ω∖S is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in Cn, then Ω={(z,w)∈X×C⁎,|w|<e−φ(z)} is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to X‾.

Goldilocks domains, a weak notion of visibility, and applications

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff–Denjoy theorems for a wide collection of domains.

Estimates of the Bergman distance on Dini-smooth bounded planar domains

Precise estimates for the Bergman distances of Dini-smooth bounded planar domains are given. These estimates imply that on such domains the Bergman distance almost coincides with the Carath\'eodory and Kobayashi distances.

Gromov (non-)hyperbolicity of certain domains in ℂN

Abstract We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for 𝒞 1 , 1 ${\mathcal{C}^{1,1}}$ -smooth convex domains in ℂ 2 ${\mathbb{C}^{2}}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in ℂ n ${\mathbb{C}^{n}}$ are given.

LIMITING BEHAVIOR OF THE KOBAYASHI DISTANCE

Given a sequence $\{D_m\}$ of bounded and convex domains in a complex Banach space, we describe the limiting behavior of the corresponding sequence $\{k_{D_m}\}$ of Kobayashi distances when the given sequence $\{D_m\}$ is either monotonic or convergent in the Hausdorff metric.

Modular data and regularity of Monge–Ampère exhaustions and of Kobayashi distance

Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge–Ampere exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite dimensional family of convex domains with squared Kobayashi distance of prescribed regularity properties. A new sharp refinement of Stoll’s characterization of \(\mathbb {C}^{n}\) is also given.

Quasi-isometries in strongly convex domains

We provide examples of quasi-isometries for strongly convex domains in Cn endowed with their Kobayashi distance.

Iteration of self-maps on a product of Hilbert balls

Let D=D1×⋯×Dp be a product of Hilbert balls, with coordinate maps πj:D¯→D¯j on the closure D¯, for j=1,…,p. Let f be a fixed-point free self-map on D, which is nonexpansive in the Kobayashi distance, and compact for p⩾2. We describe the horospheres invariant under f and show that there exist a boundary point (ξ1,…,ξp) of D and a nonempty set J⊂{1,…,p} such that each limit function h of the iterates (fn) satisfies ξj∈πj∘h(D)¯ for all j∈J and πj∘h(⋅)=ξj whenever πj∘h(D) meets the boundary of Dj. For a single Hilbert ball D1, either liminfn→∞‖f2n(0)‖<1 or (fn) converges locally uniformly to a constant map taking value at the boundary of D1.

Teichmüller distance and Kobayashi distance on subspaces of the universal Teichmüller space

It is known that the Teichmüller distance on the universal Teichmüller space T coincides with the Kobayashi distance. For a metric subspace of T having a comparable complex structure with that of T, we can similarly consider whether or not the Teichmüller distance on the subspace coincides with the Kobayashi distance. In this paper, we give a sufficient condition for metric subspaces under which the problem above has a affirmative answer. Moreover, we introduce an example of such subspaces.

Harnack inequalities, Kobayashi distances and holomorphic motions

We prove some generalizations and analogs of the Harnack inequalities for pluriharmonic, holomorphic and “almost holomorphic” functions. The results are applied to proving smoothness properties of holomorphic motions over almost complex manifolds.

Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log d_D)+C$. The lower estimate by $-1/2\log d_D+C$ is obtained under additional assumptions of $\mathcal C^1$-smoothness of a domain and a non-tangential convergence.

Totally geodesic discs in strongly convex domains

We prove that every isometry from the unit disk Δ in $${\mathbb{C}}$$ , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class $${\mathcal{C}^3}$$ in $${\mathbb{C}^n}$$ , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class $${\mathcal{C}^3}$$ in $${\mathbb{C}^n}$$ to a strongly convex bounded domain of class $${\mathcal{C}^3}$$ in $${\mathbb{C}^m}$$ , is either holomorphic or anti-holomorphic.

A Denjoy–Wolff theorem for compact holomorphic mappings in reflexive Banach spaces

Using the Kobayashi distance, we establish a Denjoy–Wolff theorem for compact holomorphic self-mappings of a bounded and strictly convex domain in a complex reflexive Banach space.

Theorems of Denjoy–Wolff type

Using the Kobayashi distance, we first establish a version of the Denjoy–Wolff theorem for a bounded and strictly convex domain in \({{\mathbb{C}}^k}\) . Next, we prove analogous results for semigroups of holomorphic mappings and the resolvents of their generators. Finally, we obtain theorems of Denjoy–Wolff type for families of holomorphic retracts of the open unit ball in a complex, reflexive, and strictly convex Banach space.

Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order \({d \in [1, +\infty]}\) . Finally, based on these results, we focus on some open problems naturally arising.

Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains

We characterize, using the Bergman kernel, Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in Cn, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing results obtained in the unit ball by Jevtić, Massaneda and Thomas, by Duren and Weir, and by MacCluer.