On highly degenerate CR maps of spheres

TL;DR

This paper classifies certain highly degenerate CR maps of spheres, revealing their structure, degree constraints, and providing new examples of rational maps with specific degeneracy properties.

Contribution

It provides a complete classification of $(N-3)$-degenerate CR maps of spheres for $N \\geq 4$, including new examples of rational maps with unique degeneracy features.

Findings

Maps have image in a 5-dimensional complex-linear space

Maps are of degree at most two or equivalent to Faran's maps

New examples of degree three rational maps with localized degeneracy

Abstract

For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is of degree at most two, or equivalent to one of the four maps into the five-dimensional sphere classified by Faran. As a byproduct of our classification we obtain new examples of rational maps of degree three which are $(N-3)$-degenerate only along a proper real subvariety and are not equivalent to polynomial maps. In particular, by changing the base point, it is possible to construct new families of nondegenerate maps.