Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings (Q1102410)

The authors obtain results concerning the continuous extension of proper holomorphic maps \(f: D_ 1\to D_ 2\) under local assumptions on the boundaries \(bD_ 1\) and \(dD_ 2\). One of the main results is the following Theorem. Let \(f: D_ 1\to D_ 2\) be a proper holomorphic map of a domain \(D_ 1\subset {\mathbb{C}}^ n\) onto a bounded domain \(D_ 2\subset {\mathbb{C}}^ n\) and let \(bD_ 1\) be of class \(C^ 2\) and strictly pseudoconvex near a point z \(0\in bD_ 1\). If there exists a sequence \(\{z^ j\}\subset D_ 1\) and that \(\lim_{j\to \infty}z^ j=z^ 0\), the limit \(\lim_{j\to \infty}f(z^ j) = w^ 0\in bD_ 2\) exists, and \(bD_ 2\) is of class \(C^ 2\) and strictly pseudoconvex near \(w^ 0\), then f extends to a Hölder continuous map with the exponent 1/2 on a neighborhood of \(z^ 0\) in \(\bar D_ 1.\) As an application, the authors prove the following result. Let \(D_ 1\subset {\mathbb{C}}^ n\) be a domain with a \(C^ 2\) plurisubharmonic defining function, and let \(D_ 2\subset \subset {\mathbb{C}}^ n\) be a bounded domain whose boundary is \(C^ 2\) strictly pseudoconvex outside a closed totally disconnected subset \(E\subset bD_ 2\). Then every proper holomorphic map of \(D_ 1\) onto \(D_ 2\) extends continuously to \(\bar D_ 1\).