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

The following result is proved: Let $D$ and $D^{\mathrm{\prime }}$ be bounded domains in ${\mathbb{C}}^n$, $\mathrm{\partial }D$ is smooth, real-analytic, simply connected, and $\mathrm{\partial }{D}^{\mathrm{\prime }}$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence $f:D\to D^{\mathrm{\prime }}$ if and only if $\mathrm{\partial }D$ and $\mathrm{\partial }{D}^{\mathrm{\prime }}$ 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.