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 ℝn are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.
"Two-point distortion theorems for harmonic mappings." Illinois J. Math. 53 (4) 1061 - 1075, Winter 2009. https://doi.org/10.1215/ijm/1290435339