On the local meromorphic extension of CR meromorphic mappings

Abstract

Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m=CRdimM\ge 1$,$n=codimM\ge 1$, $d=dimM=2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\mathcal{D}}_f,[{\Gamma }_f])$, where: 1. $f:{\mathcal{D}}_f\to Y$ is a ${\mathcal{C}}^1$-smooth mapping defined over a dense open subset ${\mathcal{D}}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure ${\Gamma }_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\mathcal{C}}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[{\Gamma }_f]=0$ in the sense of currents. We prove in this paper that $(f,{\mathcal{D}}_f,[{\Gamma }_f])$ extends meromorphically to a wedge attached to $M$ if $M$ is everywhere minimal and ${\mathcal{C}}^{\omega }$ (real analytic) or if $M$ is a ${\mathcal{C}}^{2,\alpha }$ globally minimal hypersurface.

25 pages, LaTeX. To appear in Ann. Pol. Math. 1998