Conformal and holomorphic barycenters in hyperbolic balls

TL;DR

This paper introduces conformal and holomorphic barycenters in hyperbolic balls, extending the concept of barycenters of measures on spheres to multidimensional hyperbolic spaces, highlighting differences and similarities.

Contribution

It defines and compares conformal and holomorphic barycenters in hyperbolic spaces, generalizing previous concepts from spheres to higher dimensions.

Findings

Barycenters coincide in the disk but differ in higher-dimensional balls.

The notions extend the Douady-Earle barycenter concept to hyperbolic geometry.

Differences between conformal and holomorphic barycenters are characterized in multidimensional settings.

Abstract

We introduce the notions of \textit{conformal barycenter} and \textit{holomorphic barycenter} of a measurable set $D$ in the hyperbolic ball. The two barycenters coincide in the disk, but they differ in multidimensional balls $\mathbb{C}^m \cong \mathbb{R}^{2m}$. These notions are counterparts of barycenters of measures on spheres, introduced by Douady and Earle in 1986.