# Carath\'{e}odory balls and proper holomorphic maps on multiply-connected planar domains

@inproceedings{Ng2021CaratheodoryBA, title={Carath\'\{e\}odory balls and proper holomorphic maps on multiply-connected planar domains}, author={Tuen Wai Ng and Chiu Chak Tang and Jonathan Tsai}, year={2021} }

In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carathéodory metric in some planar domains of finite connectivity greater than 2, and hence resolve a problem posed by Jarnicki, Pflug and Vigué in 1992. We also establish a corresponding result for some pseudoconvex domains in C for n ≥ 2. This result will follow from an explicit characterization (up to biholomorphisms) of proper holomorphic maps from a non-degenerate finitely-connected…

## References

SHOWING 1-10 OF 29 REFERENCES

The Structure of the Semigroup of Proper Holomorphic Mappings of a Planar Domain to the Unit Disc

- Mathematics
- 2007

Given a bounded n-connected domain Ω in the plane bounded by n non-intersecting Jordan curves and given one point bj on each boundary curve, L. Bieberbach proved that there exists a proper…

Connectedness of the Carathéodory discs for doubly connected domains

- Mathematics
- 2005

We prove that the Carathéodory discs for doubly connected domains in the complex plane are connected. Let G C be a domain and assume that no boundary component is a point. The Carathéodory distance…

On Schwarz’s Lemma in Multiply Connected Domains

- Mathematics
- 2005

Let G be a n-connected domain in the complex plane and let z0 ∈ G be fixed. We consider the class H1(G, z0) of all holomorphic functions f of G into the unit disk E with f(z0) = 0. For z ∈ Gz0 the…

Invariant distances and metrics in complex analysis

- Mathematics
- 2000

C onstructing a distance that is invariant under a given class of mappings is one of the fundamental tools for the geometric approach in mathematics. The idea goes back to Klein and even to Riemann.…

Global topology of hyperbolic components I: Cantor circle case

- Mathematics
- 2016

The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be…

Classical Topics in Complex Function Theory

- Mathematics
- 1997

An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician.…

Blaschke product for bordered surfaces

- PhysicsAnalysis and Mathematical Physics
- 2019

Any ramified holomorphic covering of a closed unit disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give two explicit constructions for a…

On a class of automorphic functions

- Mathematics
- 1902

and has shown how a certain prime form which is automorphic for the group can he expressed by a theta series. In the following article further results are given concerning this group and the…

Abel's Theorem and the allied Theory, including the Theory of the Theta Functions

- Nature
- 1897

CAYLEY'S often-quoted simile which compares the province of mathematics with “a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in…

Computing the Schottky-Klein Prime Function on the Schottky Double of Planar Domains

- Mathematics
- 2007

A numerical algorithm is presented for the computation of the Schottky-Klein prime function on the Schottky double of multiply connected circular domains in the plane. While there exist classical…