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27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
16 августа 2019 г. 14:00, г. Красноярск, Сибирский федеральный университет


A Gromov hyperbolic metric vs the hyperbolic and other related metrics

S. Sahoo



Аннотация: We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. A metric space $(D,d)$ is called Gromov hyperbolic if and only if there exist a constant $\beta>0$ such that
$$d(x,z)+d(y,w)\le (d(x,w)+d(y,z))\vee (d(x,y)+d(z,w))+2\beta $$
for all points $x,y,z,w \in D$.
For a domain $D\subsetneq \mathbb{R}^n$ equipped with the Euclidean metric, the $u_D$-metric [1] is defined by
$$ u_D(x,y)=2\log \frac{|x-y|+\max\{{\rm dist}(x,\partial D),{\rm dist}(y,\partial D)\}} {\sqrt{{\rm dist}(x,\partial D)\,{\rm dist}(y,\partial D)}}, \quad x,y\in D. $$
Ibragimov proved in [1] that the $u_D$-metric is Gromov hyperbolic and it coincides with the Vuorinen's distance ratio metric [3,4] in punctured spaces $\mathbb{R}^n\setminus\{p\}$, for $p\in\mathbb{R}^n$.
A scale invariant version of the Cassinian metric has been studied by Ibragimov in [2] which is defined by
$$ \tilde{\tau}_D(x,y)=\log\left(1+\sup_{p\in \partial D}\frac{|x-y|}{\sqrt{|x-p||p-y|}}\right), \quad x,y\in D\subsetneq \mathbb{R}^n. $$
The interesting part of this metric is that many properties in arbitrary domains are revealed in the setting of once-punctured spaces. For example, $\tilde{\tau}_D$ is a metric in an arbitrary domain $D\subsetneq \mathbb{R}^n$ if it is a metric on once-punctured spaces. The $\tilde{\tau}_D$-metric is comparable with the Vuorinen's distance ratio metric in arbitrary domains $D\subsetneq \mathbb{R}^n$ if they are comparable in the punctured spaces (see [2]).
Our purpose is to compare the $u_D$-metric with the hyperbolic and the $\tilde\tau$-metrics.
This is a joint work with Manas Ranjan Mohapatra.

Язык доклада: английский

Список литературы
  1. Zair Ibragimov, “Hyperbolizing metric spaces”, Proc. Amer. Math. Soc., 139:12 (2011), 4401–4407  crossref  mathscinet
  2. Zair Ibragimov, “A scale-invariant Cassinian metric”, J. Anal., 24:1 (2016), 111–129  crossref  mathscinet
  3. Matti Vuorinen, “Conformal invariants and quasiregular mappings”, J. Analyse Math., 45 (1985), 69–115  crossref  mathscinet
  4. Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, 1319, Springer-Verlag, Berlin, 1988 , xx+209 pp.  crossref  mathscinet


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