Bergman barycentric extensions of boundary homeomorphisms of the complex ball

We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by $$E_B(f)(z)=\text{barB}(f_{\ast }{\sigma }_z),$$ where $\text{barB}$ is the Busemann barycenter for the Bergman metric and ${\sigma }_z$ is the visual, equivalently Poisson–Szegő, measure based at $z$. We prove well-definedness, prescribed boundary values, full $\text{Aut}({\text{B}}^n)$-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 $$M^{-1}{d}_B(x,y)\le d_B(E_B(f)(x),E_B(f)(y))\le M{d}_B(x,y)\mathrm{.}$$ In contrast, for $n\ge 2$ 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.