arXiv subjects

Oliver Roth

Publications and source records attributed to Oliver Roth.

At least 37 records · Page 2Linked to original sources

arXiv2018

Universal locally univalent functions and universal conformal metrics with constant curvature

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.

math.CV↗

arXiv2018

A Convergent Star Product on the Poincaré Disc

On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra.

math.CV↗

arXiv2018

Is there a Teichmüller principle in higher dimensions?

The underlying theme of Teichmüller's papers in function theory is a general principle which asserts that every extremal problem for univalent functions of one complex variable is connected with an associated quadratic differential. The purpose of this paper is to indicate a possible way of extending Teichmüller's principle to several complex variables. This approach is based on the Loewner differential equation.

math.CV↗

arXiv2017

Support points and the Bieberbach conjecture in higher dimension

We describe some open questions related to support points in the class $S^0$ and introduce some useful techniques toward a higher dimensional Bieberbach conjecture.

math.CV↗

arXiv2016

Strong submultiplicativity of the Poincare metric

We give a direct proof of an important result of Solynin which says that the Poincaré metric is a strongly submultiplicative domain function. This result is then used to define a new capacity for compact subsets of the complex plane $\mathbb{C}$, which might be called Poincaré capacity. If the compact set $K \subseteq \mathbb{C}$ is connected, then the Poincaré capacity of $K$ is the same as the logarithmic capacity of $K$. In this special case, the submultiplicativity is well--known and can be stated as an inequality for the normalized conformal map onto the complement of $K$. Using the connection between Poincaré metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.

math.CV↗

arXiv2014

The theorems of Schottky and Landau for analytic functions omitting the n-th roots of unity

We prove sharp Landau- and Schottky-type theorems for analytic functions which omit the $n$-th roots of unity. The proofs are based on a sharp lower bound for the Poincaré metric of the complex plane punctured at the roots of unity.

math.CV↗

arXiv2014

Pontryagin's maximum principle for the Loewner equation in higher dimensions

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin's maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci and Wold we then apply our version of the Pontryagin maximum principle to obtain first--order necessary conditions for the extremal functions for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$.

math.CV↗

arXiv2014

The Schramm-Loewner equation for multiple slits

We prove that any disjoint union of finitely many simple curves in the upper half-plane can be generated in a unique way by the chordal multiple-slit Loewner equation with constant weights.

math.CV↗

arXiv2013

Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation

We describe the region $\mathcal{V}(z_0)$ of values of $f(z_0)$ for all normalized bounded univalent functions $f$ in the unit disk $\mathbb{D}$ at a fixed point $z_0 \in \mathbb{D}$. The proof is based on identifying $\mathcal{V}(z_0)$ as the reachable set of the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.

math.CV↗

arXiv2013

Composition and decomposition of indestructible Blaschke products

We prove that the composition of two indestructible Blaschke products is again an indestructible Blaschke product. We also show that if an indestructible Blaschke product is the composition of two bounded analytic functions, then both functions are indestructible Blaschke products.

math.CV↗

arXiv2013

Critical points, the Gauss curvature equation and Blaschke products

In this survey paper, we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger-Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique "maximal" Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties simliar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.

math.CV↗

arXiv2013

Maximal Blaschke Products

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari--Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.

math.CV↗

arXiv2009

Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky

An explicit formula for the generalized hyperbolic metric on the thrice--punctured sphere $¶\backslash \{z_1, z_2, z_3\}$ with singularities of order $α_j \le 1$ at $z_j$ is obtained in all possible cases $α_1+α_2+α_3 >2$. The existence and uniqueness of such a metric was proved long time ago by Picard \cite{Pic1905} and Heins \cite{Hei62}, while explicit formulas for the cases $α_1=α_2=1$ were given earlier by Agard \cite{AG} and recently by Anderson, Sugawa, Vamanamurthy and Vuorinen \cite{A}. We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel \cite{Hem79} and Minda \cite{Min87b}. As applications, sharp versions of Landau-- and Schottky--type theorems for meromorphic functions are obtained.

math.CV↗

arXiv2009

Beurling's free boundary value problem in conformal geometry

The subject of this paper is Beurling's celebrated extension of the Riemann mapping theorem \cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurling's geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling's geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincaré problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.

math.CV↗

arXiv2008

Conformal Metrics

This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.

math.CV↗

arXiv2008

On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the twodimensional curvature equation $Δu=k(z) e^{2 u}$ with $e^{2 u} \in L^1$ for bounded nonnegative curvature functions $-k(z)$ near isolated singularities is obtained.

math.AP↗

arXiv2008

A Note on Generating Functions for Hausdorff Moment Sequences

For functions $f$ whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of $w(y):=|f(γ+iy)|$ for $y>0$ ($γ\leq1$ fixed).

math.CA↗

arXiv2008

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $Δu=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $Δu=- κ(z) e^{2u}$ where $κ$ is a negative Hölder continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained.

math.AP↗