The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps

TL;DR

This paper introduces a CR analogue of the Ahlfors derivative, providing a new computable invariant for spherically equivalent CR maps that generalizes the CR Schwarzian derivative and helps distinguish sphere maps.

Contribution

It develops a CR Ahlfors derivative, extending the CR Schwarzian, and introduces a new invariant for classifying spherically equivalent CR maps.

Findings

Invariant vanishes for maps spherically equivalent to linear sphere embeddings

Invariant distinguishes many known sphere maps

Provides a computable tool for CR map classification

Abstract

We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe [23] that generalizes the CR Schwarzian derivative studied earlier by the second-named author [21]. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spherically equivalent CR maps from strictly pseudoconvex CR manifolds into a sphere. The invariant is computable and distinguishes many well-known sphere maps. In particular, it vanishes precisely when the map is spherically equivalent to the linear embedding of spheres.