Conjecture de Globevnik-Stout et theoreme de Morera pour une chaine holomorphe

Abstract

Let $D\subset \subset {\mathbb{C}}^n$ be a complex manifold of dimension $p\ge 2$ with $\C^2$ boundary in ${\mathbb{C}}^n$. Let $f$ be a $\C^1$ function on $bD$ and $V$ a generic and large enough family of complex $(n-p+1)$-planes. Let suppose that for $\nu \in V$, no connected component of $bD\cap {\mathbb{C}}_{\nu }^{n-p+1}$ is "almost" real analytic and that $f$ extends holomorphically in $D\cap {\mathbb{C}}_{\nu }^{n-p+1}$. Then $f$ extend as a holomorphic function in $D$. In a special case, this result gives a partial answer to a conjecture of Globevnik-Stout. By generalizing the theorem of Harvey-Lawson, we prove a Morera type theorem for the boundary problem in ${\mathbb{C}}^n$ which answer to a problem asked by Dolbeault and Henkin.

24 pages, LaTeX