Analytic regularity of CR maps into spheres

Let $M\subset {\mathbb C}^N$ be a connected real-analytic hypersurface and ${\mathbb S}\subset {\mathbb C}^{N'}$ the unit real sphere, $N' > N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of ${\mathbb C}^N$ and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to {\mathbb S}$ of class $\6C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in ${\mathbb C}^N$.

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Published 1 January 2003