# Finite jet determination of CR mappings - Mathematics > Complex Variables

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Abstract: We prove the following finite jet determination result for CR mappings: Givena smooth generic submanifold M of C^N, N >= 2, which is essentially finite andof finite type at each of its points, for every point p on M there exists aninteger lp, depending upper-semicontinuously on p, such that for every smoothgeneric submanifold M- of C^N of the same dimension as M, if h 1 and h 2:M,p->M- are two germs of smooth finite CR mappings with the same lp jet atp, then necessarily their k-jets agree for all positive integers k. In thehypersurface case, this result provides several new unique jet determinationproperties for holomorphic mappings at the boundary in the real-analytic case;in particular, it provides the finite jet determination of arbitraryreal-analytic CR mappings between real-analytic hypersurfaces in C^N ofD-Angelo finite type. It also yields a new boundary version of H. Cartan-suniqueness theorem: if Omega and Omega- are two bounded domains in C^N withsmooth real-analytic boundary, then there exists an integer k, depending onlyon the boundary of Omega, such that if H 1 and H 2: Omega -> Omega- are twoproper holomorphic mappings extending smoothly up to the boundary of Omega nearsome point boundary point p and agreeing up to order k at p, then necessarilyH 1=H 2.

Author: ** Bernhard Lamel, Nordine Mir**

Source: https://arxiv.org/