CR映射的有限喷流判定
Finite jet determination of CR mappings
摘要：
本文证明了CR映射的有限喷流判定结果：给定C^N，N >= 2的光滑一般子流形M，M本质上是有限的，在它的每一点上都是有限型，对M上的每一点p，存在一个上连续依赖于p的整数l（p），使得对C^N的每一个与M维数相同的光滑一般子流形M'，如果h_1和h_2：（M，p）->M'是两个在p处具有相同l（p）喷射的光滑有限CR映射芽，则它们的k-喷射必然对所有正整数k一致.在超曲面情形下，该结果提供了实解析情形下边界上全纯映射的几个新的唯一喷流判定性质，特别是，它提供了C^N中D 'Angelo有限型实解析超曲面之间任意实解析CR映射的有限喷流判定.它也产生了一个新的边界版本的H。嘉当唯一性定理：若Ω和Ω '是C^N中具有光滑实解析边界的两个有界区域，则存在一个仅依赖于Ω边界的整数k，使得如果H_1和H_2：Ω-> Ω'是两个在边界点p附近光滑地延伸到Ω边界的真全纯映射，且在p处达到k阶，则必然H_1=H_2.
英文摘要：
We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of C^N, 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 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 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 C^N 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 C^N with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary of Omega, such that if H_1 and H_2: 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 H_1=H_2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
