Two-Point Distortion Theorems for Harmonic Mappings

TL;DR

This paper extends univalence criteria for harmonic mappings to minimal surfaces, providing quantitative two-point distortion theorems and related results for curves in Euclidean space.

Contribution

It introduces new quantitative two-point distortion theorems for harmonic mappings and minimal surfaces, building on previous univalence criteria.

Findings

Developed two-point distortion theorems for harmonic mappings

Recast injectivity criteria of curves in Euclidean space quantitatively

Extended Schwarzian derivative criteria to harmonic mappings

Abstract

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.