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

Abstract

We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe that generalizes the CR Schwarzian derivative studied earlier by the second-named author. 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.Résumé. -Nous étudions un analogue CR de la dérivée au sens d'Ahlfors pour les immersions conformes de Stowe, qui généralise la dérivée schwarzienne CR étudiée antérieurement par le second auteur. Cette notion possède plusieurs propriétés importantes similaires à celles de son homologue conforme et fournit un nouvel invariant pour les applications CR, sphériquement équivalentes, de variétés CR strictement pseudoconvexes à valeurs dans la sphère. Cet invariant est calculable et permet de distinguer beaucoup d'applications CR sphériques entre elles. En particulier, il s'annule précisément quand l'application est sphériquement équivalente au plongement linéaire entre sphères.

On CR maps from the sphere into the tube over the future light cone

“…This is similar to Reiter-Son [26] and is ultimately motivated by Huang [15]. Based on an idea originated in Lamel-Son [19] and Reiter-Son [26], we introduce a tensorial invariant for CR transversal maps from a sphere or hyperquadric into the boundary of a classical domain of type IV and prove a version of Huang-Lu-Tang-Xiao [16] boundary characterization of isometric embeddings.…”

“…The notion of geometric rank is very useful in the study of sphere and hyperquadric maps, as exhibited in Huang [15] and Huang et al [16]. Moreover, it is related to the rank of the Hermitian part of the CR Ahlfors tensor of sphere and hyperquadric maps, as shown in Lamel-Son [19] and Reiter-Son [27]. Motivated by these works, we introduce the following tensor.…”

“…The first one is a recent study of the CR Ahlfors tensor and the second is the relation between the Hermitian part of the CR Ahlfors tensor and the geometric rank of sphere and hyperquadric maps. Proposition 3.3 (Lamel-Son [19]). Let M and M ′ be strictly pseudoconvex real hypersurfaces as in Definition 3.2.…”

“…The construction of the CR Ahlfors tensor is lengthy. We refer the readers to [19] for the details as we mainly work with A ′ (H), which is defined also in the case of Levi-degenerate hypersurfaces. On the other hand, this proposition suggests that the CR Ahlfors derivative is interesting only when we can choose "nice" pseudohermitian structures, which often come from a special defining functions of the hypersurfaces.…”

“…The related literature is huge and goes back to Henry Poincaré [24] in 1907. For later works, we mention for examples previous works related to CR maps of Webster [29], Faran [8,9], Forstneric [11], D'Angelo [5], Della Sala et al [6], Ebenfelt [7], Huang [15], Kim-Zaitsev [17], Lamel [19], Reiter [25] and the numerous references therein. For works on proper holomorphic maps, we refer the readers to, e.g., Alexander [1], Chan-Mok [4], Mok [21,23], Mok-Ng [22], Xiao-Yuan [31] and the references therein for more information.…”