On quasiconformal harmonic maps between surfaces

TL;DR

This paper proves that quasiconformal harmonic maps between Riemann surfaces with smooth boundaries and approximate analytic metrics are quasi-isometries under the Euclidean metric, revealing a geometric property of such mappings.

Contribution

It establishes a new link between quasiconformal harmonic maps and quasi-isometries in the setting of Riemann surfaces with smooth boundaries.

Findings

Quasiconformal harmonic maps are quasi-isometries under Euclidean metric.

The result applies to surfaces with smooth boundary and approximate analytic metrics.

Provides insight into geometric behavior of harmonic maps in complex analysis.

Abstract

It is proved the following theorem, if $w$ is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then $w$ is a quasi-isometry with respect to Euclidean metric.