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Register a new userA Gromov Hyperbolic metric and Mobius transformations
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We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for the Gromov hyperbolic metric under some families of M{o}bius transformations.
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A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $GsubsetneqRn$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $Bn$ or the upper half space $Hn$. We also obtain the sharp Lipschitz constant for a Mobius transformation $f: Grightarrow G$ between domains $G$ and $G$ in $Rn$ with respect to the metrics $v_G$ and $v_{G}$. For instance, in the case $G=G=Bn$ the result is sharp.
We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the Mobius transformations of the circle form a maximal convergence group. This completes the characterization of the Mobius group as a maximal convergence group acting on the sphere. Previously, Gehring and Martin had shown the maximality of the Mobius group on spheres of dimension greater than one. Maximality of the isometry (conformal) group of the hyperbolic plane as a uniform quasi-isometry group, uniformly quasiconformal group, and as a convergence group in which each element is topologically conjugate to an isometry may be viewed as consequences.
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 $d(z,partialOmega)lambda_Omega(z)ge1/2$ for $zinOmega,$ where $d(z,partialOmega)$ denotes the Euclidean distance from $z$ to the boundary $partialOmega.$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.
In this paper we study the global geometry of the Kobayashi metric on convex sets. We provide new examples of non-Gromov hyperbolic domains in $mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon ewline -vex, bounded and unbounded. Our first aim is to prove that if $Omega$ is a bounded weakly linearly convex domain in $mathbb{C}^n,,ngeq 2,$ and $S$ is an affine complex hyperplane intersecting $Omega,$ then the domain $Omegasetminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $Omegasetminus S,$ where $S$ is relatively closed in $Omega$ and $Omega$ is a convex domain, depends only on that how $S$ looks near the boundary, i.e., whether $S$ near $partialOmega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.
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