A Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces

TL;DR

This paper establishes a Schwarz lemma for harmonic mappings between unit balls in real Euclidean spaces, providing bounds on the image of smaller balls under such mappings, extending complex plane results.

Contribution

It introduces a Schwarz lemma for harmonic mappings in real Euclidean spaces, generalizing previous complex plane results.

Findings

Harmonic mappings between unit balls can be controlled within smaller balls.

The result extends known complex plane Schwarz lemmas to higher-dimensional real spaces.

Provides a new tool for analyzing harmonic mappings in Euclidean spaces.

Abstract

In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.