Proper holomorphic maps from the unit disk to some unit ball

D’Angelo, John; Huo, Zhenghui; Xiao, Ming

doi:10.1090/proc/13425

Proper holomorphic maps from the unit disk to some unit ball
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by John P. D’Angelo, Zhenghui Huo and Ming Xiao PDF

Proc. Amer. Math. Soc. 145 (2017), 2649-2660 Request permission

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

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, DOI 10.1353/ajm.2011.0044
J. A. Cima and T. J. Suffridge, Boundary behavior of rational proper maps, Duke Math. J. 60 (1990), no. 1, 135–138. MR 1047119, DOI 10.1215/S0012-7094-90-06004-1
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
John P. D’Angelo, Proper holomorphic mappings, positivity conditions, and isometric imbedding, J. Korean Math. Soc. 40 (2003), no. 3, 341–371. MR 1973906, DOI 10.4134/JKMS.2003.40.3.341
John P. D’Angelo and Jiří Lebl, Homotopy equivalence for proper holomorphic mappings, Adv. Math. 286 (2016), 160–180. MR 3415683, DOI 10.1016/j.aim.2015.09.007
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, DOI 10.1080/17476930902759403
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
Franc Forstnerič, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), no. 1, 31–61. MR 969413, DOI 10.1007/BF01394144
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
Xiaojun Huang and Shanyu Ji, Mapping $\mathbf B^n$ into $\mathbf B^{2n-1}$, Invent. Math. 145 (2001), no. 2, 219–250. MR 1872546, DOI 10.1007/s002220100140
ShanYu Ji and Yuan Zhang, Classification of rational holomorphic maps from $\Bbb B^2$ into $\Bbb B^{\Bbb N}$ with degree 2, Sci. China Ser. A 52 (2009), no. 12, 2647–2667. MR 2577180, DOI 10.1007/s11425-009-0147-y
Jiří Lebl, Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics, Michigan Math. J. 60 (2011), no. 3, 603–628. MR 2861091, DOI 10.1307/mmj/1320763051