Finite jet determination of CR mappings
Abstract
We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of CN, N >= 2, which is essentially finite and of finite type at each of its points, for every point p on M there exists an integer l(p), depending upper-semicontinuously on p, such that for every smooth generic submanifold M' of CN of the same dimension as M, if h1 and h2: (M,p)->M' are two germs of smooth finite CR mappings with the same l(p) jet at p, then necessarily their k-jets agree 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 Omega and Omega' are two bounded domains in CN with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary of Omega, such that if H1 and H2: Omega -> Omega' are two proper holomorphic mappings extending smoothly up to the boundary of Omega near some point boundary point p and agreeing up to order k at p, then necessarily H1=H2.