Conformal and holomorphic barycenters in hyperbolic balls

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.

15 pages