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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.20662 |
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| _version_ | 1866910239692947456 |
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| author | Cao, Wensheng Ge, Zhijian |
| author_facet | Cao, Wensheng Ge, Zhijian |
| contents | We extend the notion of conformal barycenter, recently introduced by
Jačimović 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
$\int_D Φ_c(q)\, \dLam(q) = \mathbf{0}$, where
$Φ_c$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conformal Barycenters in Quaternionic Hyperbolic Balls Cao, Wensheng Ge, Zhijian Mathematical Physics 51M10, 53C35, 15B33 We extend the notion of conformal barycenter, recently introduced by Jačimović 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 $\int_D Φ_c(q)\, \dLam(q) = \mathbf{0}$, where $Φ_c$ 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. |
| title | Conformal Barycenters in Quaternionic Hyperbolic Balls |
| topic | Mathematical Physics 51M10, 53C35, 15B33 |
| url | https://arxiv.org/abs/2605.20662 |