Universal Locally Univalent Functions and Universal Conformal Metrics with Constant Curvature

Universal Locally Univalent Functions and Universal Conformal Metrics with Constant Curvature Constr Approx https://doi.org/10.1007/s00365-018-9434-6 Universal Locally Univalent Functions and Universal Conformal Metrics with Constant Curvature 1 1 Daniel Pohl · Oliver Roth Received: 29 September 2017 / Revised: 21 February 2018 / Accepted: 3 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature. Keywords Universal functions · Runge theory · Locally univalent functions · Conformal metrics · Constant curvature Mathematics Subject Classification 30E10 · 30F45 · 30K20 · 30K99 1 Introduction Let  be a domain in the complex plane C, and let H() be the space of all holomorphic functions on . We think of H() as a (closed) subspace of the Fréchet space C () of all complex-valued continuous functions on  equipped with the compact-open topology of locally uniform convergence. We denote by Aut () the group of all Communicated by Edward B. Saff. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Constructive Approximation Springer Journals

Universal Locally Univalent Functions and Universal Conformal Metrics with Constant Curvature

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Numerical Analysis; Analysis
ISSN
0176-4276
eISSN
1432-0940
D.O.I.
10.1007/s00365-018-9434-6
Publisher site
See Article on Publisher Site

Abstract

Constr Approx https://doi.org/10.1007/s00365-018-9434-6 Universal Locally Univalent Functions and Universal Conformal Metrics with Constant Curvature 1 1 Daniel Pohl · Oliver Roth Received: 29 September 2017 / Revised: 21 February 2018 / Accepted: 3 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature. Keywords Universal functions · Runge theory · Locally univalent functions · Conformal metrics · Constant curvature Mathematics Subject Classification 30E10 · 30F45 · 30K20 · 30K99 1 Introduction Let  be a domain in the complex plane C, and let H() be the space of all holomorphic functions on . We think of H() as a (closed) subspace of the Fréchet space C () of all complex-valued continuous functions on  equipped with the compact-open topology of locally uniform convergence. We denote by Aut () the group of all Communicated by Edward B. Saff.

Journal

Constructive ApproximationSpringer Journals

Published: May 31, 2018

References

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