Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Proper holomorphic maps from the unit disk to some unit ball


Authors: John P. D’Angelo, Zhenghui Huo and Ming Xiao
Journal: Proc. Amer. Math. Soc. 145 (2017), 2649-2660
MSC (2010): Primary 32H35, 51F25, 32M99; Secondary 30J99
DOI: https://doi.org/10.1090/proc/13425
Published electronically: December 8, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $ 2$.


References [Enhancements On Off] (What's this?)

  • [1] M. Salah Baouendi, Peter Ebenfelt, and Xiaojun Huang, Holomorphic mappings between hyperquadrics with small signature difference, Amer. J. Math. 133 (2011), no. 6, 1633-1661. MR 2863372, https://doi.org/10.1353/ajm.2011.0044
  • [2] J. A. Cima and T. J. Suffridge, Boundary behavior of rational proper maps, Duke Math. J. 60 (1990), no. 1, 135-138. MR 1047119, https://doi.org/10.1215/S0012-7094-90-06004-1
  • [3] John P. D'Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR 1224231
  • [4] John P. D'Angelo, Proper holomorphic mappings, positivity conditions, and isometric imbedding, J. Korean Math. Soc. 40 (2003), no. 3, 341-371. MR 1973906, https://doi.org/10.4134/JKMS.2003.40.3.341
  • [5] John P. D'Angelo and Jiří Lebl, Homotopy equivalence for proper holomorphic mappings, Adv. Math. 286 (2016), 160-180. MR 3415683, https://doi.org/10.1016/j.aim.2015.09.007
  • [6] John P. D'Angelo and Jiří Lebl, On the complexity of proper holomorphic mappings between balls, Complex Var. Elliptic Equ. 54 (2009), no. 3-4, 187-204. MR 2513534, https://doi.org/10.1080/17476930902759403
  • [7] Peter Ebenfelt, Xiaojun Huang, and Dmitri Zaitsev, The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics, Amer. J. Math. 127 (2005), no. 1, 169-191. MR 2115664
  • [8] Franc Forstnerič, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), no. 1, 31-61. MR 969413, https://doi.org/10.1007/BF01394144
  • [9] Xiaojun Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions, J. Differential Geom. 51 (1999), no. 1, 13-33. MR 1703603
  • [10] Xiaojun Huang and Shanyu Ji, Mapping $ \mathbf {B}^n$ into $ \mathbf {B}^{2n-1}$, Invent. Math. 145 (2001), no. 2, 219-250. MR 1872546, https://doi.org/10.1007/s002220100140
  • [11] ShanYu Ji and Yuan Zhang, Classification of rational holomorphic maps from $ \mathbb{B}^2$ into $ \mathbb{B}^{\mathbb{N}}$ with degree 2, Sci. China Ser. A 52 (2009), no. 12, 2647-2667. MR 2577180, https://doi.org/10.1007/s11425-009-0147-y
  • [12] Jiří Lebl, Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics, Michigan Math. J. 60 (2011), no. 3, 603-628. MR 2861091, https://doi.org/10.1307/mmj/1320763051

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 32H35, 51F25, 32M99, 30J99

Retrieve articles in all journals with MSC (2010): 32H35, 51F25, 32M99, 30J99


Additional Information

John P. D’Angelo
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: jpda@math.uiuc.edu

Zhenghui Huo
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130-4899
Email: huo@wustl.edu

Ming Xiao
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: mingxiao@illinois.edu

DOI: https://doi.org/10.1090/proc/13425
Keywords: Proper holomorphic mappings, unit disk, unit ball, unitary equivalence, homotopy equivalence
Received by editor(s): June 6, 2016
Received by editor(s) in revised form: August 1, 2016
Published electronically: December 8, 2016
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society