CR manifolds admitting a CR contraction

TL;DR

This paper classifies smooth CR manifolds with a CR contraction, showing they embed into complex space as polynomial hypersurfaces determined by resonances, and extends the contraction to a holomorphic polynomial normal form.

Contribution

It provides a complete classification of CR manifolds admitting a CR contraction and describes their embedding and normal form in complex space.

Findings

CR manifolds with a CR contraction are embedded as polynomial hypersurfaces

The contraction extends to a holomorphic polynomial normal form

Results connect CR geometry with polynomial normal forms and resonances

Abstract

We classify the germs of $\mathcal{C}^\infty$ CR manifolds that admit a smooth CR contraction. We show that such a CR manifold is embedded into $\CC^n$ as a real hypersurface defined by a polynomial defining function consisting of monomials whose degrees are completely determined by the extended resonances of the contraction. Furthermore the contraction map extends to a holomorphic contraction that coincides in fact with its polynomial normal form. Consequently, several results concerning Complex and CR geometry are derived.