Corpus ID: 237940499

Continuity and bi-Lipschitz properties of the Hurwitz and its invariant metrics

  title={Continuity and bi-Lipschitz properties of the Hurwitz and its invariant metrics},
  author={Arstu and Swadesh Kumar Sahoo},
  • Arstu, Swadesh Kumar Sahoo
  • Published 25 September 2021
  • Mathematics
The classical Poincaré’s hyperbolic metric was first introduced in the early nineties. Since then, a family of conformal metrics that are closely related to the hyperbolic metric was introduced by several mathematicians. Just to name a few, they are the Hurwitz metric [13], the Gardiner-Lakic metric [3], the Hahn metric [6], the quasihyperbolic metric [4], and many more. As the hyperbolic metric is valid for only hyperbolic domains of the plane due to its very difficult nature to compute… Expand


Comaparing Poincaré densities
  • Ann. of Math., 154
  • 2001
A generalized Hurwitz metric.
In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincare's hyperbolic metric when theExpand
Carathéodory Density of the Hurwitz Metric on Plane Domains
It is well-known that the Caratheodory metric is a natural generalization of the Poincare metric, namely, the hyperbolic metric of the unit disk. In 2016, the Hurwitz metric was introduced by D.Expand
Boundary Behaviour of some Conformal Invariants on Planar Domains
The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumann–Carathéodory rigidity constant, theExpand
The Hurwitz Metric
In 1904 Hurwitz considered the extremal problem of maximizing $$\left| f'(0) \right| $$f′(0) over all holomorphic functions in $${\mathbb {D}}$$D such that $$f(0)=0$$f(0)=0 and $$f(z) \neExpand
The rate of convergence of the hyperbolic density on sequences of domains
It is known that if a sequence of domains $U_n$ converges to a domain $U$ in the Caratheodory sense then the hyperbolic densities on $U_n$ converge to the hyperbolic density on $U$. In this paper, weExpand
The Geometry of Complex Domains
Preface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 TheExpand
Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric
We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.
Hyperbolic geometry from a local viewpoint
Introduction 1. Elementary transformations 2 Hyperbolic metric in the unit disk 3. Holomorphic functions 4. Topology and uniformization 5. Discontinuous groups 6 Fuchsian groups 7. General hyperbolicExpand
Some inequalities for the Poincaré metric of plane domains
Abstract.In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain InExpand