Related papers: Hyperbolic domains in real Euclidean spaces

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to…

In this paper we study the hyperbolicity in the sense of Gromov of domains in $\mathbb{R}^d$ $(d\geq3)$ with respect to the minimal metric introduced by Forstneri\v{c} and Kalaj. In particular, we prove that every bounded strongly minimally…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already…

Complete hyperbolicity of small Euclidean balls with respect to a C^1-smooth almost complex structure standard at origin is improved to give a complete hyperbolicity of strictly pseudoconvex domains. More precise (and lower) regularity…

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in $\mathbb{C}^2$, which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.

In this work we solve a couple of well known open problems related to the quasihyperbolic metric. In the case of planar domains, our first main result states that quasihyperbolic geodesics are unique in simply connected domains. As the…

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in…

It is well known that a hyperbolic domain in the complex plane has uniformly perfect boundary precisely when the product of its hyperbolic density and the distance function to its boundary has a positive lower bound. We extend this…

We prove that for a pseudoconvex domain of the form $\mathfrak{A} = \{(z, w) \in \mathbb C^2 : v > F(z, u)\}$, where $w = u + iv$ and F is a continuous function on ${\mathbb C}_z \times {\mathbb R}_u$, the following conditions are…

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for…

Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type…

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For…

In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex…

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved…

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however,…

Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect,…

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then…

Let $\Omega$ be a domain in $\mathbb{C}$ with hyperbolic metric $\lambda_\Omega(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $\Omega$ is (Euclidean) convex if and only if…