Finite jet determination of CR mappings

Bernhard Lamel, Nordine Mir

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)153-177
Number of pages25
JournalAdvances in Mathematics
Volume216
Issue number1
DOIs
Publication statusPublished - 1 Dec 2007

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Keywords

  • CR mapping
  • Finite jet determination

ASJC Scopus subject areas

  • Mathematics(all)

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