### Abstract

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M ⊂ C^{N}, N ≥ 2, that is essentially finite and of finite type at each of its points, for every point p ∈ M there exists an integer ℓ_{p}, depending upper-semicontinuously on p, such that for every smooth generic submanifold M^{′} ⊂ C^{N} of the same dimension as M, if h_{1}, h_{2} : (M, p) → M^{′} are two germs of smooth finite CR mappings with the same ℓ_{p} jet at p, then necessarily j_{p}^{k} h_{1} = j_{p}^{k} h_{2} for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C^{N} of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω, Ω^{′} ⊂ C^{N} are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if H_{1}, H_{2} : Ω → Ω^{′} are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p ∈ ∂ Ω and agreeing up to order k at p, then necessarily H_{1} = H_{2}.

Original language | English |
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Pages (from-to) | 153-177 |

Number of pages | 25 |

Journal | Advances in Mathematics |

Volume | 216 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Dec 2007 |

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### Keywords

- CR mapping
- Finite jet determination

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*216*(1), 153-177. https://doi.org/10.1016/j.aim.2007.05.007