Finite jet determination of CR mappings
Résumé
We prove the following finite jet determination result for CR mappings : Given a smooth generic submanifold $M\subset \C^N$, $N\geq 2$, that is essentially finite and of finite type at each of its points, for every point $p\in M$ there exists an integer $\ell_p$, depending upper-semicontinuously on $p$, such that for every smooth generic submanifold $M'\subset \CN$ of the same dimension as that of $M$, if $h_1,h_2\colon (M,p)\to M'$ are two germs of smooth finite CR mappings with the same $\ell_p$ jet at $p$, then necessarily $j^k_ph_1=j_p^kh_2$ for all positive integers $k$. Specified to the hypersurface case, this result already provides several new unique jet determination properties for holomorphic mappings at the boundary even in the real-analytic case. Among other things, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces of $\CN$ of D'Angelo finite type. It also yields the following new boundary version of H. Cartan's uniqueness theorem: if $\Omega,\Omega'\subset \CN$ are two bounded domains with smooth real-analytic boundary then there exists an integer $k$, depending only on the boundary $\partial \Omega$, such that if $H_1,H_2\colon \Omega \to \Omega'$ are two proper holomorphic mappings extending smoothly up to $\partial \Omega$ near some point $p\in\partial \Omega$ and agreeing up to order $k$ at $p$, then necessarily $H_1=H_2$.
Domaines
Variables complexes [math.CV]
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