# Geometric theory of meromorphic functions

@inproceedings{Eremenko2002GeometricTO, title={Geometric theory of meromorphic functions}, author={Alexandre Eremenko}, year={2002} }

This is a survey of results on the following problem. Let X be a simply connected Riemann surface spread over the Riemann sphere. How are the properties of the uniformizing function of this surface related to the geometric properties of the surface? 2000 Mathematics Subject Classication: 30D30, 30D35, 30D45, 30F45.

## 15 Citations

Critical values of generating functions of totally positive sequences

Parametrization of various classes of entire and meromorphic functions by critical values is interesting from the point of view of geometric theory of meromorphic functions [5, 10, 16] and is also…

Log-Riemann Surfaces

- Mathematics
- 2015

We introduce the notion of log-Riemann surfaces. These are Riemann surfaces given by cutting and pasting planes together isometrically, and come equipped with a holomorphic local diffeomorphism to C…

Painleve I, Coverings of the Sphere and Belyi Functions

- Mathematics, Physics
- 2014

The theory of poles of solutions of Painleve I (PI) is equivalent to the Nevanlinna problem of constructing a meromorphic function ramified over five points—counting multiplicities—and without…

Entire functions arising from trees

- MathematicsScience China Mathematics
- 2021

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function $f$ with only two critical values $\pm 1$ and no asymptotic values such that…

Comb functions

- Mathematics
- 2011

We discuss a class of regions and conformal mappings which are useful in several problems of approximation theory, harmonic analysis and spectral theory.1 MSC: 30C20, 41A10, 47B36, 41A50.

Complex flows, escape to infinity and a question of Rubel

- Mathematics
- 2021

Let f be a transcendental entire function. It was shown in a previous paper [12] that the holomorphic flow ż = f(z) always has infinitely many trajectories tending to infinity in finite time. It will…

The order conjecture fails in S

- Mathematics
- 2015

We construct an entire function f with only three singular values, whose order can change under a quasiconformal equivalence.

The Schwarzian derivative and the Wiman-Valiron property

- Mathematics
- 2013

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has…

Resurgent deformation quantisation

- Physics, Mathematics
- 2014

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. The algebra would be large enough to capture quantum effects that escape ordinary formal…

Analytic Continuation of Eigenvalues of a Quartic Oscillator

- Mathematics, Physics
- 2009

We consider the Schrödinger operator on the real line with even quartic potential x4 + αx2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of…

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Critical values of generating functions of totally positive sequences

Parametrization of various classes of entire and meromorphic functions by critical values is interesting from the point of view of geometric theory of meromorphic functions [5, 10, 16] and is also…

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Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of…

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I n the nineteenth century the theory of analytic functions grew into a major field of mathematical research. Mostly, mathematicians were interested in properties of specific classes of analytic…

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Let f be an entire function of finite order, real on the real axis, and possessing only real zeros. A classical problem, proposed by G. P6lya [9], [10] and A. Wiman [1], is to determine, from the…