Caratheodory Metric Research Articles

Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carath�odory metric

We construct a counter example to the Besicovitch covering property for some class of Carnot groups equipped with their Carnot - Caratheodory metric. The construction uses some regularity assumptions for minimal geodesics and for the distance function to the origin. As an example we show that groups of type H, thus including all Heisenberg groups, satisfy these assumptions. We prove in particular that, in groups of type H, the distance function to the origin is Pansu-differentiable outside the center with continuous Pansu-differential there.

Stability of Mappings with Bounded Distortion on the Heisenberg Group

A stability theorem is proved for mappings with bounded distortion over John domains in the Heisenberg group furnished with the Carnot–Caratheodory metric.

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Caratheodory metrics, the singular Caratheodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂn are studied.

Nontangential Weighted Limit of the Infinitesimal Carathéodory Metric in an h-Extendible Boundary Point of a Smooth Bounded Pseudoconvex Domain in Cn

A boundary point of a domain in Cn is said to be h-extendible if its Catlin's multitype coincides with its D'Angelo's type. The main purpose of this paper is to study the existence of nontangential weighted limit of the infinitesimal Caratheodory metric in such a point of a smooth bounded pseudoconvex domain in Cn.

Smoothness of kobayashi metric of ellipsoids

Lempert proved that the Kobayashi metric and the Caratheodory metric of smooth strongly convex domains are smooth away from the zero section of the holomorphic tangent bundle. We will show that if the strong convexity is replaced by the weaker condition of strict convexity the above smoothness result is no longer valid even if the boundary is real analytic. We study the smoothness of the Kobayashi metric of ellipsoids. We prove that the Kobayashi metric of domains of the form , where m≥3/2, is piecewise C 3 off the zero section, but not C 3.

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.

Bounded analytic sets in Banach spaces

Conditions are given which enable or disable a complex space X to be mapped biholomorphically onto a bounded closed analytic subset of a Banach space. They involve on the one hand the Radon-Nikodym property and on the other hand the completeness of the Caratheodory metric of X.

A characterization of the polydisc

The Kobayashi metric of a complex manifold dominates its Caratheodory metric. It thus seems natural to investigate cases of equality of these metrics, especially so since they correspond to the extremal case in Schwarz's lemma. The open unit polydisc A" in ~n is one of the simplest cases in which these metrics coincide, and the following theorem shows how these metrics characterize the complex structure of the polydisc. (Definitions of the metrics and the terms "finitely compact" and "indicatrix" appear in Sect. 2.)

The Caratheodory Metric and its Majorant Metrics

One of the main purposes of the present paper is to provide a proof for the following statement:Theorem A. Let M be a complex manifold of a complex dimension n. Let be a fixed point in M such that there exists a square integrable holomorphic n-form a(z) on M with .

The Carathéodory metric of the annulus

An explicit formula is given for the Caratheodory metric of the annulus in the complex plane. The infinitesimal Caratheodory metric, which is just the norm of the differentiation at a point regarded as a functional on the algebra of holomorphic functions, is also determined.