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 \[ E_B(f)(z)=\barB(f_*σ_z), \] 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 \[ M^{-1}d_B(x,y)\le d_B(E_B(f)(x),E_B(f)(y))\le M d_B(x,y). \] In contrast, for \(n\ge2\) 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.