### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### 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

**Finite jet determination of CR mappings.** / Lamel, Bernhard; Mir, Nordine.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 216, no. 1, pp. 153-177. https://doi.org/10.1016/j.aim.2007.05.007

}

TY - JOUR

T1 - Finite jet determination of CR mappings

AU - Lamel, Bernhard

AU - Mir, Nordine

PY - 2007/12/1

Y1 - 2007/12/1

N2 - We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M ⊂ CN, N ≥ 2, that is essentially finite and of finite type at each of its points, for every point p ∈ M there exists an integer ℓp, depending upper-semicontinuously on p, such that for every smooth generic submanifold M′ ⊂ CN of the same dimension as M, if h1, h2 : (M, p) → M′ are two germs of smooth finite CR mappings with the same ℓp jet at p, then necessarily jpk h1 = jpk h2 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 CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω, Ω′ ⊂ CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if H1, H2 : 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 H1 = H2.

AB - We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M ⊂ CN, N ≥ 2, that is essentially finite and of finite type at each of its points, for every point p ∈ M there exists an integer ℓp, depending upper-semicontinuously on p, such that for every smooth generic submanifold M′ ⊂ CN of the same dimension as M, if h1, h2 : (M, p) → M′ are two germs of smooth finite CR mappings with the same ℓp jet at p, then necessarily jpk h1 = jpk h2 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 CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω, Ω′ ⊂ CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if H1, H2 : 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 H1 = H2.

KW - CR mapping

KW - Finite jet determination

UR - http://www.scopus.com/inward/record.url?scp=34548209153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548209153&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2007.05.007

DO - 10.1016/j.aim.2007.05.007

M3 - Article

VL - 216

SP - 153

EP - 177

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -