Bergman barycentric extensions of boundary homeomorphisms of the complex ball
Abstract
We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by EB(f)(z)=barB(f*σz), where barB is the Busemann barycenter for the Bergman metric and σz is the visual, equivalently Poisson--Szegő, measure based at z. We prove well-definedness, prescribed boundary values, full Aut(Bn)-naturality, and interior real-analyticity. For every CR-quasisymmetric boundary homeomorphism, the Bergman barycentric extension is a quasi-isometry of complex hyperbolic space; moreover EB(f−1) is a coarse inverse of EB(f). Under sufficiently small positive CR cross-ratio distortion we obtain a sharper Tukia-type theorem: for every M>1, the extension is a real-analytic diffeomorphism satisfying M−1dB(x,y)≤ dB(EB(f)(x),EB(f)(y))≤ M dB(x,y). In contrast, for n≥2 there are smooth CR-orientation-preserving CR-quasisymmetric boundary diffeomorphisms whose barycentric extensions are non-injective and, after normalization, have singular differential. Thus large-scale quasi-isometric control persists on the full CR-quasisymmetric class, whereas local non-degeneracy requires stronger boundary control.
Related
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BibTeX
@misc{kalaj2026bergman,
title = {{Bergman barycentric extensions of boundary homeomorphisms of the complex ball}},
author = {David Kalaj and Anton Gjokaj and Vladimir Jacimovic},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28014},
eprint = {2609.28014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28014},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bergman barycentric extensions of boundary homeomorphisms of the complex ball
AU - David Kalaj
AU - Anton Gjokaj
AU - Vladimir Jacimovic
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28014
UR - https://arxiv.org/abs/2609.28014
AB - We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by E_B(f)(z)=barB(f_*σ_z), where barB is the Busemann barycenter for the Bergman metric and σ_z is the visual, equivalently Poisson--Szegő, measure based at z. We prove well-definedness, prescribed boundary values, full Aut(B^n)-naturality, and interior real-analyticity. For every CR-quasisymmetric boundary homeomorphism, the Bergman barycentric extension is a quasi-isometry of complex hyperbolic space; moreover E_B(f^−1) is a coarse inverse of E_B(f). Under sufficiently small positive CR cross-ratio distortion we obtain a sharper Tukia-type theorem: for every M>1, the extension is a real-analytic diffeomorphism satisfying M^−1d_B(x,y)≤ d_B(E_B(f)(x),E_B(f)(y))≤ M d_B(x,y). In contrast, for n≥2 there are smooth CR-orientation-preserving CR-quasisymmetric boundary diffeomorphisms whose barycentric extensions are non-injective and, after normalization, have singular differential. Thus large-scale quasi-isometric control persists on the full CR-quasisymmetric class, whereas local non-degeneracy requires stronger boundary control.
ER -