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.
Constructive Approximation – Springer Journals
Published: May 31, 2018
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