On CR maps between hyperquadrics and Winkelmann hypersurfaces

TL;DR

This paper investigates the conditions under which CR maps from hyperquadrics to Winkelmann hypersurfaces extend to local holomorphic isometric embeddings, linking the extension to the vanishing of the Hermitian part of the CR Ahlfors derivative.

Contribution

It establishes a new criterion connecting the extension of CR maps to the vanishing of the Hermitian part of the CR Ahlfors derivative, generalizing previous results.

Findings

CR maps extend to local holomorphic isometric embeddings if Hermitian part of CR Ahlfors derivative vanishes.

The proof relates geometric rank of CR maps to their CR Ahlfors derivative.

Extension criterion applies to maps into hyperquadrics and Winkelmann hypersurfaces.

Abstract

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel-Son and a recent result of Huang-Lu-Tang-Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain K\"ahler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative.