The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2137-2167.

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.

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.

Online First:
Published online:
DOI: 10.5802/aif.3438
Classification: 32V05,  32H35,  53A30
Keywords: Sphere maps, Ahlfors derivative, CR maps
Lamel, Bernhard 1; Son, Duong Ngoc 2

1 Texas A & M University Qatar, Science Program, Education City, Doha, Qatar
2 Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
License: CC-BY-ND 4.0
     author = {Lamel, Bernhard and Son, Duong Ngoc},
     title = {The {CR} {Ahlfors} derivative and a new invariant for spherically equivalent {CR} maps},
     journal = {Annales de l'Institut Fourier},
     pages = {2137--2167},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {5},
     year = {2021},
     doi = {10.5802/aif.3438},
     language = {en},
     url = {}
TI  - The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps
JO  - Annales de l'Institut Fourier
PY  - 2021
DA  - 2021///
SP  - 2137
EP  - 2167
VL  - 71
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  -
UR  -
DO  - 10.5802/aif.3438
LA  - en
ID  - AIF_2021__71_5_2137_0
ER  - 
%0 Journal Article
%T The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps
%J Annales de l'Institut Fourier
%D 2021
%P 2137-2167
%V 71
%N 5
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.3438
%G en
%F AIF_2021__71_5_2137_0
Lamel, Bernhard; Son, Duong Ngoc. The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2137-2167. doi : 10.5802/aif.3438.

[1] Ahlfors, Lars V. Cross-ratios and Schwarzian derivatives in R n , Complex analysis, Edited by J.Hersch, A. Huber, articles dedicated to Albert Pfluger on the occasion of his 80th birthday, Birkhäuser, 1988, pp. 1-15 | Zbl

[2] Bedford, Eric The Dirichlet problem for some overdetermined systems on the unit ball in n , Pac. J. Math., Volume 51 (1974) no. 1, pp. 19-25 | DOI | MR | Zbl

[3] D’Angelo, John P. Proper holomorphic maps between balls of different dimensions, Mich. Math. J., Volume 35 (1988) no. 1, pp. 83-90 | MR | Zbl

[4] D’Angelo, John P. Polynomial proper holomorphic mappings between balls II., Mich. Math. J., Volume 38 (1991) no. 1, pp. 53-65 | MR | Zbl

[5] D’Angelo, John P. Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, 8, CRC Press, 1993 | Zbl

[6] D’Angelo, John P.; Lebl, Jiří Homotopy equivalence for proper holomorphic mappings, Adv. Math., Volume 286 (2016), pp. 160-180 | DOI | MR | Zbl

[7] D’Angelo, John P.; Xiao, Ming Symmetries in CR complexity theory, Adv. Math., Volume 313 (2017), pp. 590-627 | DOI | MR | Zbl

[8] Dragomir, Sorin On pseudohermitian immersions between strictly pseudoconvex CR manifolds, Am. J. Math., Volume 117 (1995) no. 1, pp. 169-202 | DOI | Zbl

[9] Ebenfelt, Peter; Huang, Xiaojun; Zaitsev, Dmitri Rigidity of CR-immersions into Spheres, Commun. Anal. Geom., Volume 12 (2004) no. 3, pp. 631-670 | DOI | MR | Zbl

[10] Ebenfelt, Peter; Minor, André On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions, Trans. Am. Math. Soc., Volume 366 (2014) no. 11, pp. 5693-5706 | DOI | MR | Zbl

[11] Faran, James; Huang, Xiaojun; Ji, Shanyu; Zhang, Yuan Rational and polynomial maps between balls, Pure Appl. Math. Q., Volume 6 (2010) no. 3, pp. 829-842 | DOI | Zbl

[12] Graham, C. Robin; Lee, John M. Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J., Volume 57 (1988) no. 3, pp. 697-720 | MR | Zbl

[13] Ji, Shanyu; Yuan, Yuan Flatness of CR submanifolds in a sphere, Sci. China, Math., Volume 53 (2010) no. 3, pp. 701-718 | MR | Zbl

[14] Lee, John M. The Fefferman metric and pseudo-Hermitian invariants, Trans. Am. Math. Soc., Volume 296 (1986) no. 1, pp. 411-429 | MR | Zbl

[15] Lee, John M. Pseudo-Einstein Structures on CR Manifolds, Am. J. Math., Volume 110 (1988) no. 1, pp. 157-178 | MR | Zbl

[16] Lee, John M.; Melrose, Richard B. Boundary behavior of the complex Monge–Ampère equation, Acta Math., Volume 148 (1982) no. 1, pp. 159-192 | Zbl

[17] Li, Song-Ying; Luk, Hing-Sun An explicit formula for the Webster pseudo-Ricci curvature on real hypersurfaces and its application for characterizing balls in n , Commun. Anal. Geom., Volume 14 (2006) no. 4, pp. 673-701 | MR | Zbl

[18] Li, Song-Ying; Son, Duong Ngoc The Webster scalar curvature and sharp upper and lower bounds for the first positive eigenvalue of the Kohn–Laplacian on real hypersurfaces, Acta Math. Sin., Engl. Ser., Volume 34 (2018) no. 8, pp. 1248-1258 | MR | Zbl

[19] Osgood, Brad; Stowe, Dennis The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J., Volume 67 (1992) no. 1, pp. 57-99 | MR | Zbl

[20] Rudin, Walter Function theory in the unit ball of Cn, Grundlehren der Mathematischen Wissenschaften, 241, Springer, 1980 | DOI | Zbl

[21] Son, Duong Ngoc The Schwarzian derivative and Möbius equation on strictly pseudoconvex CR manifolds, Commun. Anal. Geom., Volume 26 (2018) no. 2, pp. 237-269 | MR | Zbl

[22] Son, Duong Ngoc Semi-isometric CR immersions of CR manifolds into Kähler manifolds and applications (2019) ( to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze)

[23] Stowe, Dennis An Ahlfors derivative for conformal immersions, J. Geom. Anal., Volume 25 (2015) no. 1, pp. 592-615 | DOI | MR | Zbl

[24] Tanaka, Noboru A differential geometric study on strongly pseudoconvex manifolds, Lectures in Mathematics. Department of Mathematics, Kyoto University, 9, Kinokuniya Company Ltd., Tokyo, 1975 | MR | Zbl

[25] Watanabe, Chikara On proper monomial mappings from 2-ball B 2 , to 4-ball B 4 , Ann. Sci. Kanazawa Univ., Volume 29 (1992), pp. 9-13 | MR

[26] Webster, Sidney M. The rigidity of C-R hypersurfaces in a sphere, Indiana Univ. Math. J., Volume 28 (1979) no. 3, pp. 405-416 | DOI | MR | Zbl

Cited by Sources: