Conformal Barycenters in Quaternionic Hyperbolic Balls

TL;DR

This paper generalizes the concept of conformal barycenters from complex to quaternionic hyperbolic balls, establishing existence, uniqueness, and invariance properties through convexity analysis.

Contribution

It extends the conformal barycenter notion to quaternionic hyperbolic spaces, proving key properties and providing explicit examples.

Findings

Existence and uniqueness of quaternionic conformal barycenters.

Invariance of barycenters under quaternionic hyperbolic isometries.

Explicit examples of barycenters for finite point sets.

Abstract

We extend the notion of conformal barycenter, recently introduced by Ja\v{c}imovi\'{c} and Kalaj for the complex hyperbolic ball, to the quaternionic unit ball $\BH$. The quaternionic conformal barycenter of a measurable set $D$ with finite hyperbolic measure and finite first moment is defined as the unique point $c$ such that $\int_D \Phi_c(q)\, \dLam(q) = \mathbf{0}$, where $\Phi_c$ is the quaternionic Hua involution exchanging $0$ and $c$. Equivalently, it is the unique minimum of the energy functional $G(x) = \int_D \log\cosh^2\!\big(\frac12 d_H(x,y)\big)\, \dLam(y)$. We prove existence and uniqueness using the strict geodesic convexity of $G$, which is established by a direct computation along geodesics. The barycenter is invariant under the full isometry group $\mathrm{Sp}(n,1)$. We also treat finite point sets and provide explicit examples.