Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in ${\mathbb{C}}^n$

Abstract

The following result is proved: Let $D$ and $D^{\prime }$ be bounded domains in ${\mathbb{C}}^n$, $\partial D$ is smooth, real-analytic, simply connected, and $\partial D^{\prime }$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence $f:D\to D^{\prime }$ if and only if $\partial D$ and $\partial D^{\prime }$ a locally CR-equivalent. This leads to a characterization of the equivalence relationship between bounded domains in ${\mathbb{C}}^n$ modulo proper holomorphic correspondences in terms of CR-equivalence of their boundaries.

17 pages

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