AI 中文总结
本文定义复球边界同胚的Bergman重心延拓，证明其良定义性、边界值、自然性与实解析性，并表明CR拟对称同胚的延拓为拟等距，小失真下为满足双Lipschitz界的实解析微分同胚，但光滑情形可能失去局部非退化性。
AI 中文摘要
我们定义了复单位球边界同胚的Bergman重心延拓，形式为\\[ E_B(f)(z)=\barB(f_*σ_z), \\] 其中\\(\barB\\)是Bergman度量的Busemann重心，\\(σ_z\\)是基于\\(z\\)的视觉测度，等价于Poisson–Szegő测度。我们证明了该延拓的良定义性、给定的边界值、完全的\\(\Aut(\B^n)\\)-自然性以及内部实解析性。对于每个CR拟对称边界同胚，Bergman重心延拓是复双曲空间的拟等距；此外，\\(E_B(f^{-1})\\)是\\(E_B(f)\\)的粗逆。在足够小的正CR交叉比失真下，我们获得了更精细的Tukia型定理：对于每个\\(M>1\\)，该延拓是满足\\[ M^{-1}d_B(x,y)\le d_B(E_B(f)(x),E_B(f)(y))\le M d_B(x,y) \\] 的实解析微分同胚。相比之下，对于\\(n\ge2\\)，存在光滑的保持CR方向的CR拟对称边界微分同胚，其重心延拓是非单射的，并且在归一化后具有奇异微分。因此，大尺度拟等距控制在完整的CR拟对称类上持续成立，而局部非退化性需要更强的边界控制。
英文摘要
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.
