A note on CR mappings of positive codimension

TL;DR

This paper proves an approximation theorem for smooth CR mappings between real-analytic CR submanifolds, showing that such mappings can be approximated by real-analytic CR mappings with matching jets at generic points.

Contribution

It establishes an Artin type approximation result for CR mappings of positive codimension, extending the understanding of their local approximation properties.

Findings

Dense open subset of points with approximable CR mappings

Existence of real-analytic CR mappings matching jets of smooth mappings

Approximation holds for arbitrary jet order k

Abstract

We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k.