Boundaries of Analytic Varieties

Abstract

We prove that every smooth CR manifold $M\subset\subset \C^n$, of hypersurface type, has a complex strip-manifold extension in $\C^n$. If $M$ is, in addition, pseudoconvex-oriented, it is the "exterior" boundary of the strip. In turn, the strip extends to a variety with boundary $M$ (Rothstein-Sperling Theorem); in case $M$ is contained in a pseudoconvex boundary with no complex tangencies, the variety is embedded in $\C^n$. Altogether we get: $M$ is the boundary of a variety (Harvey-Lawson Theorem); if $M$ is pseudoconvex oriented the singularities of the variety are isolated in the interior; if $M$ lies in a pseudoconvex boundary, the variety is embedded in $\C^n$ (and is still smooth at $M$)