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Une charactérisation des quadriques hermitiennes dans \(\mathbb{C}^ n\). (A characterization of Hermite quadrics in \(\mathbb{C}^ n\)). (French) Zbl 0747.35007

Main result: Let \(S:=\{r(z)=0\}\) be a connected hypersurface in \(\mathbb{C}^ n\) of class \({\mathcal C}^ 8\). The following conditions are equivalent
(i) If \(r(z)=0\) and \(\sum^ n_{j=1}(\partial r/\partial z_ j)(z)w_ j=0\) then \(\sum^ n_{j,k=1}(\partial^ 2r/\partial z_ j\partial z_ k)(z)w_ jw_ k=0\);
(ii) Either \(S\) is contained in a Hermite quadric, or \(S\) can be foliated by complex hyperplanes.
Reviewer: J.Siciak (Kraków)

MSC:

35J25 Boundary value problems for second-order elliptic equations
32V40 Real submanifolds in complex manifolds
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
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References:

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