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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 \(U\).

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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