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Finite jet determination of holomorphic mappings at the boundary. (English) Zbl 1015.32031
A classical theorem of H. Cartan states that an automorphism $$f$$ of a bounded domain $$D\subset\mathbb C^N$$ is completely determined by its 1-jet, i.e. its value and derivatives of order one, at any point $$Z_0\in D$$. If $$D$$, in addition, is assumed to be $$C^\infty$$-smooth bounded and strictly pseudoconvex, then by Fefferman’s theorem any such automorphism extends smoothly to the boundary $$\partial D$$ as an automorphism $$\partial D\to\partial D$$. It is then natural to ask: is $$f$$ completely determined by a finite jet at a boundary point $$p\in\partial D$$?
The author proves the following Theorem: Let $$M,M'\subset\mathbb C^N$$ be $$C^\infty$$-smooth real hypersurfaces, and $$U\subset\mathbb C^N$$ be an open connected subset with $$M$$ in its boundary. Let $$f, g: U\to\mathbb C^N$$ be holomorphic mappings which extend smoothly to $$M$$ and send $$M$$ diffeomorphically into $$M'$$. If $$M$$ is $$k_0$$-nondegenerate at a point $$p_0\in M$$ and $$(\partial^\alpha_zf)(p_0) = (\partial^\alpha_zg)(p_0)$$, $$\forall \alpha \in\mathbb Z_+^N: |\alpha|\leq 2k_0,$$ then $$f\equiv g$$ in $$U$$.

##### MSC:
 32V40 Real submanifolds in complex manifolds 32H99 Holomorphic mappings and correspondences 32V25 Extension of functions and other analytic objects from CR manifolds
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