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arXiv · 2605.20662

Conformal Barycenters in Quaternionic Hyperbolic Balls

Wensheng Cao·Zhijian Ge
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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.

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Wensheng Cao, Zhijian Ge. 2026-05-20. Conformal Barycenters in Quaternionic Hyperbolic Balls. https://arxiv.org/abs/2605.20662

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