Finite jet determination of CR mappings
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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 p, 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 p 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 , CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary , such that if H1, H2 : are two proper holomorphic mappings extending smoothly up to near some point p and agreeing up to order k at p, then necessarily H1 = H2. 2007 Elsevier Inc. All rights reserved.
