Abstract
We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Notes
D’Angelo uses \(1-\lambda z_1 z_2\), which we put into our normal form.
References
Alexander, H.: Proper holomorphic mappings in \(C^{n}\). Indiana Univ. Math. J. 26(1), 137–146 (1977). https://doi.org/10.1512/iumj.1977.26.26010. (0022-2518)
Catlin, D.W., D’Angelo, J.P.: A stabilization theorem for Hermitian forms and applications to holomorphic mappings. Math. Res. Lett. 3(2), 149–166 (1996). https://doi.org/10.4310/MRL.1996.v3.n2.a2. (1073-2780)
Chiappari, S.A.: Holomorphic extension of proper meromorphic mappings. Michigan Math. J. 38(2), 167–174 (1991). https://doi.org/10.1307/mmj/1029004326. (0026-2285)
Cima, J.A., Suffridge, T.J.: Boundary behavior of rational proper maps. Duke Math. J. 60(1), 135–138 (1990). https://doi.org/10.1215/S0012-7094-90-06004-1. (0012-7094)
Courant, R.: Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Appendix by M. Schiffer, p. xiii+330. Interscience Publishers Inc, New York (1950)
D’Angelo, J.P.: Rational CR maps
D’Angelo, J.P.: Several complex variables and the geometry of real hypersurfaces, p. xiv+272. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993). (0-8493-8272-6)
D’Angelo, J.P.: Hermitian Analysis, Cornerstones, From Fourier Series to Cauchy–Riemann Geometry; Second edition of [MR3134931]. Birkhäuser/Springer, Cham (2019) (978-3-030-16513-0, 978-3-030-16514-7). https://doi.org/10.1007/978-3-030-16514-7
D’Angelo, J.P.: Rational Sphere Maps, Progress in Mathematics, 341, Birkhäuser/Springer, Cham, (2021) (\(\copyright \) 2021, xiii+233). https://doi.org/10.1007/978-3-030-75809-7
D’Angelo, J.P., Lebl, J.: Homotopy equivalence for proper holomorphic mappings. Adv. Math. 286, 160–180 (2016). https://doi.org/10.1016/j.aim.2015.09.007. (0001–8708)
D’Angelo, J.P., Xiao, M.: Symmetries in CR complexity theory. Adv. Math. 313, 590–627 (2017). https://doi.org/10.1016/j.aim.2017.04.014. (0001–8708)
D’Angelo, J.P., Huo, Z., Xiao, M.: Proper holomorphic maps from the unit disk to some unit ball. Proc. Am. Math. Soc. 145(6), 2649–2660 (2017). https://doi.org/10.1090/proc/13425. (0002-9939)
Dor, A.: Proper holomorphic maps between balls in one co-dimension. Ark. Mat. 28(1), 49–100 (1990). https://doi.org/10.1007/BF02387366. (0004-2080)
Ebenfelt, P.: Partial rigidity of degenerate CR embeddings into spheres. Adv. Math. 239, 72–96 (2013). https://doi.org/10.1016/j.aim.2013.02.011. (0001–8708)
Faran, J.J.: Maps from the two-ball to the three-ball. Invent. Math. 68(3), 441–475 (1982). https://doi.org/10.1007/BF01389412. (0020-9910)
Faran, J.J.: The linearity of proper holomorphic maps between balls in the low codimension case. J. Differ. Geom. 24(1), 15–17 (1986). (0022-040X)
Faran, J., Huang, X., Ji, S., Zhang, Y.: Polynomial and rational maps between balls. Pure Appl. Math. Q. 6, 3 ((2010)). (Special Issue: In honor of Joseph J. Kohn., 829–842, 1558-8599). https://doi.org/10.4310/PAMQ.2010.v6.n3.a10
Forstnerič, F.: Extending proper holomorphic mappings of positive codimension. Invent. Math. 95(1), 31–61 (1989). https://doi.org/10.1007/BF01394144. (0020-9910)
Hamada, H.: Rational proper holomorphic maps from \({ B}^n\) into \({ B}^{2n}\). Math. Ann. 331(3), 693–711 (2005). https://doi.org/10.1007/s00208-004-0606-2. (0025-5831)
Horn, R., Johnson, A., Charles, R.: Matrix Analysis, vol. 2, p. xviii+643. Cambridge University Press, Cambridge (2013)
Huang, X.: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. J. Differ. Geom. 51(1), 13–33 (1999). (0022-040X)
Huang, X., Ji, S., Xu, D.: A new gap phenomenon for proper holomorphic mappings from \(B^n\) into \(B^N\). Math. Res. Lett. 13(4), 515–529 (2006). https://doi.org/10.4310/MRL.2006.v13.n4.a2.(1073-2780)
Huang, X., Ji, S., Yin, W.: On the third gap for proper holomorphic maps between balls. Math. Ann. 358, 1–2 (2014). https://doi.org/10.1007/s00208-013-0952-z. (115-142)
Lebl, J.: Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics. Michigan Math. J. 60(3), 603–628 (2011). https://doi.org/10.1307/mmj/1320763051. (0026-2285)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was in part supported by Simons Foundation collaboration Grant 710294.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
