On the local meromorphic extension of CR meromorphic mappings

Abstract

Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [\Gamma_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ 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 ${\cal C}^1$-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 $(f,{\cal D}_f, [\Gamma_f])$ extends meromorphically to a wedge attached to $M$ if $M$ is everywhere minimal and ${\cal C}^{\omega}$ (real analytic) or if $M$ is a ${\cal C}^{2,\alpha}$ globally minimal hypersurface.