Conformal and holomorphic barycenters in hyperbolic balls

Abstract

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