arXiv subjects

Peter Duren

Publications and source records attributed to Peter Duren.

4 recordsLinked to original sources

arXiv2014

Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts

We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper lifts of a class of harmonic mappings defined in the unit disk. The extension is based on fibrations of space by circles in domain and image that correspond to each other in a natural way. Convexity plays an essential role in the analysis. As a corollary we derive a sufficient condition for the underlying harmonic mapping to be univalent in the disk, with an explicit quasiconformal extension to the extended plane that generalizes the well known formula by Ahlfors-Weill.

math.CV↗

arXiv2010

Two-Point Distortion Theorems for Harmonic Mappings

In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ${\Bbb R}^n$ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.

math.CV↗

arXiv2010

Ahlfors-Weill Extensions for a Class of Minimal Surfaces

The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions. The extension is obtained by a reflection across the boundary of the surface using a family of Euclidean circles orthogonal to the surface. This gives a geometric generalization of the Ahlfors-Weill formula and extends the minimal surface. Thus one obtains a homeomorphism of $\overline{\mathbb{C}}$ onto a toplological sphere in $\overline{\mathbb{R}^3} = \mathbb{R}^3 \cup \{\infty\}$ that is real-analytic off the boundary. The hypotheses involve bounds on a generalized Schwarzian derivative for harmonic mappings in term of the hyperbolic metric of the disk and the Gaussian curvature of the minimal surface. Hyperbolic convexity plays a crucial role.

math.CV↗

arXiv2007

Schwarzian Derivatives and Uniform Local Univalence

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.

math.CV↗