Finite jet determination of CR mappings

TL;DR

This paper establishes finite jet determination results for CR mappings on generic submanifolds, providing new uniqueness properties and boundary extension results for holomorphic mappings in complex analysis.

Contribution

It proves finite jet determination for CR mappings on essentially finite, finite type submanifolds, with applications to boundary uniqueness and holomorphic extension.

Findings

Finite jet determination for CR mappings on generic submanifolds.

New boundary uniqueness theorems for holomorphic mappings.

Finite jet determination of real-analytic CR mappings between hypersurfaces.

Abstract

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.