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Classification of Holomorphic Mappings of Hyperquadrics from \(\mathbb {C}^2\) to \(\mathbb {C}^3\)

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Abstract

We give a new proof of Faran’s and Lebl’s results by means of a new CR-geometric approach and classify all holomorphic mappings from the sphere in \(\mathbb {C}^2\) to Levi-nondegenerate hyperquadrics in \(\mathbb {C}^3\). We use the tools developed by Lamel, which allow us to isolate and study the most interesting class of holomorphic mappings. This family of so-called nondegenerate and transversal maps we denote by \(\mathcal {F}\). For \(\mathcal {F}\) we introduce a subclass \(\mathcal {N}\) of maps that are normalized with respect to the group \(\mathcal {G}\) of automorphisms fixing a given point. With the techniques introduced by Baouendi–Ebenfelt–Rothschild and Lamel we classify all maps in \(\mathcal {N}\). This intermediate result is crucial to obtain a complete classification of \(\mathcal {F}\) by considering the transitive part of the automorphism group of the hyperquadrics.

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References

  1. Alexander, H.: Holomorphic mappings from the ball and polydisc. Math. Ann. 209, 249–256 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexander, H.: Proper holomorphic mappings in \(\mathbb{C}^{n}\). Indiana Univ. Math. J. 26(1), 137–146 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Parametrization of local biholomorphisms of real analytic hypersurfaces. Asian J. Math. 1(1), 1–16 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton University Press, Princeton (1999)

    MATH  Google Scholar 

  5. Baouendi, M.S., Ebenfelt, P.: Local geometric properties of real submanifolds in complex space. Bull. Am. Math. Soc. (N.S.) 37(3), 309–336 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Transversality of holomorphic mappings between real hypersurfaces in different dimensions. Commun. Anal. Geom. 15(3), 589–611 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1(4), 333–354 (1932)

    MathSciNet  MATH  Google Scholar 

  8. Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. Pura Appl. 11(1), 17–90 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z., Ji, S., Dekang, X.: Rational holomorphic maps from \(\mathbb{B}^2\) into \(\mathbb{B}^N\) with degree \(2\). Sci. China A 49(11), 1504–1522 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cima, J.A., Suffridge, T.J.: A reflection principle with applications to proper holomorphic mappings. Math. Ann. 265(4), 489–500 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cima, J.A., Suffridge, T.J.: Proper holomorphic mappings from the two-ball to the three-ball. Trans. Am. Math. Soc. 311(1), 227–239 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. D’Angelo, J.P.: Proper holomorphic maps between balls of different dimensions. Michigan Math. J. 35(1), 83–90 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)

    Google Scholar 

  15. Ebenfelt, P., Rothschild, L.P.: Transversality of CR mappings. Am. J. Math. 128(5), 1313–1343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ebenfelt, P., Shroff, R.: Partial rigidity of CR embeddings of real hypersurfaces into hyperquadrics with small signature difference. ArXiv e-prints (2010)

  17. Faran, J.J.: Maps from the two-ball to the three-ball. Invent. Math. 68(3), 441–475 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Faran, J.J.: The linearity of proper holomorphic maps between balls in the low codimension case. J. Differ. Geom. 24(1), 15–17 (1986)

    MathSciNet  MATH  Google Scholar 

  19. Forstnerič, F.: Extending proper holomorphic mappings of positive codimension. Invent. Math. 95(1), 31–61 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang, X., Ji, S.: Mapping \(\mathbb{B}^n\) into \(\mathbb{B}^{2n-1}\). Invent. Math. 145(2), 219–250 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, X., Ji, S., Dekang, X.: A new gap phenomenon for proper holomorphic mappings from \(\mathbb{B}^n\) into \(\mathbb{B}^N\). Math. Res. Lett. 13(4), 515–529 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Huang, X.: On a semi-rigidity property for holomorphic maps. Asian J. Math. 7(4), 463–492 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ji, S.: A new proof for Faran’s theorem on maps between \(\mathbb{B}^2\) and \(\mathbb{B}^3\). Recent Advances in Geometric Analysis. Volume 11 of Advanced Lectures in Mathematics (ALM), pp. 101–127. International Press, Somerville (2010)

  25. Lamel, B.: Holomorphic maps of real submanifolds in complex spaces of different dimensions. Pac. J. Math. 201(2), 357–387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lebl, J.: Hermitian Forms Meet Several Complex Variables-Minicourse on CR Geometry Using Hermitian Forms (2011). http://www.jirka.org/scv-mini/scv-mini.pdf. Accessed 2 Sept 2014

  27. Lebl, J.: Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics. Michigan Math. J. 60(3), 603–628 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Meylan, F.: Degree of a holomorphic map between unit balls from \({\mathbb{C}}^2\) to \({\mathbb{C}}^n\). Proc. Am. Math. Soc. 134, 1023–1030 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pinčuk, S.I.: On proper holomorphic mappings of strictly pseudoconvex domains. Sib. Math. J. 15, 644–649 (1974). doi:10.1007/BF00967440

    Article  MATH  Google Scholar 

  30. Poincaré, M.H.: Les fonctions analytiques de deux variables et la représentation conforme. Rendiconti del Circolo Matematico di Palermo 1884–1940(23), 185–220 (1907). doi:10.1007/BF03013518

    Article  MATH  Google Scholar 

  31. Reiter, M.: Holomorphic mappings of hyperquadrics from \({\mathbb{C}}^2\) to C3. PhD Thesis, University of Vienna (2014). http://othes.univie.ac.at/33603

  32. Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of \(n\) complex variables. J. Math. Soc. Jpn. 14, 397–429 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  33. Webster, S.M.: On mapping an \(n\)-ball into an \((n+1)\)-ball in complex spaces. Pacific J. Math. 81(1), 267–272 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wolfram Research: Mathematica 7.0.1.0. Wolfram Research Inc, Champaign (2008)

Download references

Acknowledgments

The author would like to thank Bernhard Lamel, Giuseppe della Sala and Nordine Mir for their interest in this work and many discussions related to the topic of this article. The author was supported by the FWF, Projects Y377 and I382, and QNRF, Project NPRP 7-511-1-098.

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Authors and Affiliations

  1. Texas A&M University at Qatar, PO Box 23874, Doha, Qatar

    Michael Reiter

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  1. Michael Reiter
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Correspondence to Michael Reiter.

Appendices

Appendix 1: Formula for Jet Parameterization

In Lemma 4.3 we have the following formulas: Denote \(\Psi =(f_1,f_2,g)\). We order the monomials by degree and by assigning the weight \(1\) to \(z\) and the weight \(2\) to the variable \(\chi \). The numerator of \(f_1(z,2 {{\mathrm{\mathrm {i}}}}z \chi )\) is the following expression:

$$\begin{aligned}&2 {{\mathrm{\varepsilon }}}z + 6 A_2 z \chi + {{\mathrm{\mathrm {i}}}}C_{22} z^3 + 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21} z^2 \chi + 6 {{\mathrm{\varepsilon }}}B_2 z \chi ^2+ \Bigl (2 {{\mathrm{\varepsilon }}}A_2 + A_{22} - C_{13} \Bigr ) z^3 \chi \\&\quad - 2 \Bigl (3 {{\mathrm{\mathrm {i}}}}A_3 + 3 {{\mathrm{\varepsilon }}}B_{12} + A_2 (7 {{\mathrm{\varepsilon }}}- 3 {{\mathrm{\mathrm {i}}}}B_{21}) \Bigr ) z^2 \chi ^2+ 2 A_2 B_2 z \chi ^3 \\&\quad + \Bigl (6 A_2^2 + {{\mathrm{\mathrm {i}}}}A_{13} + {{\mathrm{\varepsilon }}}(-1 - 2 B_{21}^2+ B_{22} + C_3) - {{\mathrm{\mathrm {i}}}}C_4 - 2 {{\mathrm{\mathrm {i}}}}B_2 C_{22}\Bigr ) z^3 \chi ^2 \\&\quad - 2 \Bigl (5 A_2^2 + 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 + 4 A_2 B_{12} + B_2 (6 - 2 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21})\Bigr ) z^2 \chi ^3 \\&\quad + \Bigl (-A_4 - 2 A_{22} B_2 + B_{12} + 2 A_3 B_{21} + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}(4 A_3 + B_{13} - 4 B_{12} B_{21}) \\&\quad + A_2 (5 + 4 {{\mathrm{\varepsilon }}}B_2 - 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21} - 2 B_{21}^2+ B_{22} + 3 C_3) + 2 B_2 C_{13}\Bigr ) z^3 \chi ^3 \\&\quad + 2 {{\mathrm{\mathrm {i}}}}\Bigl (B_2 (4 A_3 + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{12}) + A_2 \bigl (-5 B_3 + B_2 (5 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}+ B_{21})\bigr )\Bigr ) z^2 \chi ^4 \\&\quad + \Bigl (2 {{\mathrm{\mathrm {i}}}}B_3 + 2 {{\mathrm{\mathrm {i}}}}A_3 B_{12} + {{\mathrm{\mathrm {i}}}}A_2 \bigl (4 A_3 + B_{13} + B_{12} (-6 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}- 4 B_{21})\bigr )\\&\quad + 2 A_2^2 (5 {{\mathrm{\varepsilon }}}- 2 B_2 - {{\mathrm{\mathrm {i}}}}B_{21}) \\&\quad + {{\mathrm{\varepsilon }}}(-B_4 + 2 B_{12}^2 + 4 B_3 B_{21}) + B_2 \bigl (-2 {{\mathrm{\mathrm {i}}}}A_{13} - 6 {{\mathrm{\mathrm {i}}}}B_{21} \\&\quad + {{\mathrm{\varepsilon }}}(2 - B_{22} + 2 C_3) + 2 {{\mathrm{\mathrm {i}}}}C_4\bigr )+ {{\mathrm{\mathrm {i}}}}B_2^2C_{22} \Bigr ) z^3 \chi ^4- 2 A_2^2 B_2 z^2 \chi ^5 \\&\quad + \Bigl (4 A_2^3 + 2 A_4 B_2 + A_{22} B_2^2+ 3 A_2^2 B_{12} + 5 B_2 B_{12} + 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 B_{12} - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_2 B_{13} \\&\quad - 2 A_3 \bigl (B_3 + B_2 ({{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}+ B_{21}) \bigr ) - A_2 \bigl (6 {{\mathrm{\varepsilon }}}B_2^2+ B_4 - 2 B_{12}^2 + B_3 (-8 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}- 4 B_{21}) \\&\quad + B_2 (-4 + 8 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21} + B_{22} + 2 C_3)\bigr ) - B_2^2 C_{13}\Bigr ) z^3 \chi ^5 \\&\quad - 2 {{\mathrm{\mathrm {i}}}}B_2 \Bigl (A_3 B_2 - A_2 B_3\Bigr ) z^2 \chi ^6 \\&\quad + \Bigl (-2 {{\mathrm{\varepsilon }}}B_3^2 + B_2 (4 {{\mathrm{\mathrm {i}}}}B_3 + {{\mathrm{\varepsilon }}}B_4 - 2 {{\mathrm{\mathrm {i}}}}A_3 B_{12}) \\&\quad - {{\mathrm{\mathrm {i}}}}A_2 \bigl (2 A_3 B_2 - 4 B_3 B_{12} + B_2(6 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{12} + B_{13})\bigr ) \\&\quad + 2 A_2^2 \bigl (-B_2^2+ 2 {{\mathrm{\mathrm {i}}}}B_3 + B_2 ({{\mathrm{\varepsilon }}}- 2 {{\mathrm{\mathrm {i}}}}B_{21})\bigr ) + B_2^2(3 {{\mathrm{\varepsilon }}}+ {{\mathrm{\mathrm {i}}}}A_{13}- 3 {{\mathrm{\varepsilon }}}C_3- {{\mathrm{\mathrm {i}}}}C_4)\Bigr ) z^3 \chi ^6 \\&\quad + \Bigl (B_2 \bigl (-A_4 B_2 + 2 A_3 (-{{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_2 + B_3)\bigr ) + 3 A_2^2 B_2 B_{12}\\&\quad + A_2 \bigl (-2 B_3^2 + B_2 (4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 + B_4) \\&\quad - B_2^2(-3 + C_3) \bigr )\Bigr ) z^3 \chi ^7 + 2 {{\mathrm{\mathrm {i}}}}A_2 B_2 \Bigl (-A_3 B_2 + A_2 B_3\Bigr ) z^3 \chi ^8 \end{aligned}$$

The numerator of \(f_2(z,2 {{\mathrm{\mathrm {i}}}}z \chi )\) is equal to the following formula:

$$\begin{aligned}&2 {{\mathrm{\varepsilon }}}z^2 + 2 A_2 z^3 + 6 A_2 z^2 \chi + \Bigl (-1 + C_3\Bigr ) z^3 \chi + 4 {{\mathrm{\varepsilon }}}B_2 z^2 \chi ^2\\&\quad - \Bigl (2 {{\mathrm{\mathrm {i}}}}A_3 + 6 A_2 ({{\mathrm{\varepsilon }}}+ B_2) + {{\mathrm{\varepsilon }}}B_{12}\Bigr ) z^3 \chi ^2 \\&\quad - 4 A_2 B_2 z^2 \chi ^3 + \Bigl (-4 A_2^2 - 2 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 - A_2 B_{12} + B_2 \left( 1 + 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21} - 3 C_3\right) \Bigr ) z^3 \chi ^3\\&\quad - 6 {{\mathrm{\varepsilon }}}B_2^2 z^2 \chi ^4 + \Bigl (2 {{\mathrm{\mathrm {i}}}}B_2 (A_3 + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{12}) \!+\! A_2 \bigl (6 B_2^2- 2 {{\mathrm{\mathrm {i}}}}B_3 \!+\! 4 B_2 ({{\mathrm{\varepsilon }}}+ {{\mathrm{\mathrm {i}}}}B_{21})\bigr )\Bigr ) z^3 \chi ^4\\&\quad - 2 A_2 B_2^2 z^2 \chi ^5 + B_2 \Bigl (4 A_2^2 - 2 A_2 B_{12} + B_2 (-3 - 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{21} + 3 C_3)\Bigr ) z^3 \chi ^5\\&\quad + B_2^2\Bigl (2 {{\mathrm{\mathrm {i}}}}A_3 + 3 {{\mathrm{\varepsilon }}}B_{12} + 2 A_2 ({{\mathrm{\varepsilon }}}- B_2 - 2 {{\mathrm{\mathrm {i}}}}B_{21})\Bigr ) z^3 \chi ^6 + B_2^2\Bigl (2 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 + 3 A_2 B_{12} \\&\quad - B_2 (-3 + C_3)\Bigr ) z^3 \chi ^7 - 2 {{\mathrm{\mathrm {i}}}}B_2^2\Bigl (A_3 B_2 - A_2 B_3\Bigr ) z^3 \chi ^8 \end{aligned}$$

The numerator of \(g(z,2 {{\mathrm{\mathrm {i}}}}z \chi )\) is equal to the following formula:

$$\begin{aligned}&4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}z \chi + 12 {{\mathrm{\mathrm {i}}}}A_2 z \chi ^2 - 2 C_{22} z^3 \chi + \Bigl (4 {{\mathrm{\mathrm {i}}}}- 8 {{\mathrm{\varepsilon }}}B_{21}\Bigr ) z^2 \chi ^2 + 12 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_2 z \chi ^3 \\&\quad + 2 {{\mathrm{\mathrm {i}}}}\Bigl (4 {{\mathrm{\varepsilon }}}A_2 + A_{22} - C_{13}\Bigr ) z^3 \chi ^2 + 12 \Bigl (A_3 - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{12} + A_2 (-{{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}- B_{21})\Bigl ) z^2 \chi ^3 \\&\quad + 4 {{\mathrm{\mathrm {i}}}}A_2 B_2 z \chi ^4\\&\quad + 2 \Bigl (8 {{\mathrm{\mathrm {i}}}}A_2^2 - A_{13} - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}(2 + 2 B_{21}^2- B_{22} - 2 C_3) + C_4 + 2 B_2 C_{22}\Bigr ) z^3 \chi ^3 \\&\quad - 4 \Bigl (2 {{\mathrm{\mathrm {i}}}}A_2^2 - 4 {{\mathrm{\varepsilon }}}B_3 + 4 {{\mathrm{\mathrm {i}}}}A_2 B_{12} + B_2 (3 {{\mathrm{\mathrm {i}}}}+ 2 {{\mathrm{\varepsilon }}}B_{21}) \Bigr ) z^2 \chi ^4 \\&\quad + 2 \Bigl (-{{\mathrm{\mathrm {i}}}}A_4 - 2 {{\mathrm{\mathrm {i}}}}A_{22} B_2 - {{\mathrm{\varepsilon }}}B_{13}- 2 A_3 ({{\mathrm{\varepsilon }}}- {{\mathrm{\mathrm {i}}}}B_{21}) + 4 {{\mathrm{\varepsilon }}}B_{12} B_{21} \\&\quad + A_2 \bigl (4 {{\mathrm{\varepsilon }}}B_{21} - 2 {{\mathrm{\mathrm {i}}}}B_{21}^2 + {{\mathrm{\mathrm {i}}}}(-2 + B_{22} + 4 C_3) \bigr ) + 2 {{\mathrm{\mathrm {i}}}}B_2 C_{13}\Bigr ) z^3 \chi ^4 \\&\quad - 4 \Bigl (B_2 (4 A_3 + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_{12}) + A_2 \bigl (-5 B_3 + B_2 ({{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}+ B_{21}) \bigr )\Bigr ) z^2 \chi ^5 \\&\quad + 2 \Bigl (-2 A_3 B_{12} - A_2 \bigl (2 A_3 + B_{13} + B_{12} (-4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}- 4 B_{21}) \bigr ) + 2 A_2^2 (-4 {{\mathrm{\mathrm {i}}}}B_2 + B_{21}) \\&\quad - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}\bigl (B_4 - 2 (B_{12}^2 + 2 B_3 B_{21})\bigr ) + B_2 \bigl (2 A_{13} + 2 B_{21} - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}(-2 + B_{22}) - 2 C_4 \bigr ) \\&\quad - B_2^2 C_{22} \Bigr ) z^3 \chi ^5- 2 {{\mathrm{\mathrm {i}}}}\Bigl (-2 A_4 B_2 - A_{22} B_2^2- 2 A_2^2 B_{12} - 2 B_2 B_{12} - 4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_3 B_{12} \\&\quad + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}B_2 B_{13} + 2 A_3 \bigl (B_3+ B_2 ({{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}+ B_{21})\bigr )\\&\quad + A_2 \bigl (4 {{\mathrm{\varepsilon }}}B_2^2+ B_4 - 2 B_{12}^2 + B_3 (-4 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}- 4 B_{21}) + B_2 (-2 + B_{22} + 4 C_3)\bigr ) \\&\quad + B_2^2 C_{13} \Bigr ) z^3 \chi ^6 + 4 B_2 \Bigl (A_3 B_2 - A_2 B_3\Bigr ) z^2 \chi ^7 \\&\quad - 2 \Bigl (A_{13} B_2^2+ 2 A_2^2 B_3 + 2 B_2 B_3 - 2 A_3 B_2 B_{12} - A_2 (2 A_3 B_2 - 4 B_3 B_{12} + B_2 B_{13}) \\&\quad + {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}(2 B_3^2 - B_2 B_4 + 2 B_2^2C_3) - B_2^2 C_4\Bigr ) z^3 \chi ^7\\&\quad - 2 {{\mathrm{\mathrm {i}}}}\Bigl (A_4 B_2^2- 2 A_3 B_2 B_3 + A_2 (2 B_3^2 - B_2 B_4)\Bigr ) z^3 \chi ^8 \end{aligned}$$

The denominator of \(H\) is of the following form:

Appendix 2: Case A and B

In the proof of Lemma 4.5 Figs. 5 and 6 occur:

Fig. 5
figure 5

Diagram for Case A

Full size image
Fig. 6
figure 6

Diagram for Case B

Full size image

Appendix 3: Formulas for \({\psi }_k\) and \(\widehat{\psi }_k\)

In Lemma 4.5 we have the following formulas:

We have \(\widehat{\psi }_k = \psi _k\) for \(k=3,4,5\).

Appendix 4: Standard Parameters

Here we give the standard parameters needed in the proofs of Theorem 1.5 given in Sect. 5.2 and Lemma 6.2. First we list the standard parameters for \(H_1\), the renormalization of \(\mathcal {H}_{1}^{{{\mathrm{\varepsilon }}}}\).

$$\begin{aligned} R_1 :=&\sqrt{\frac{1 + 6 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4}{1 + 2 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4}}, \quad c_{1 1}' := \frac{ c_1 (1 - 4 {{\mathrm{\varepsilon }}}r_0^2 - r_0^4) - 2 {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}r_0 \lambda _1}{\lambda _1 (1+ 6{{\mathrm{\varepsilon }}}r_0^2 + r_0^4)}, \\ c_{2 1}' :=&\frac{2 r_0 (2 c_1 - {{\mathrm{\mathrm {i}}}}{{\mathrm{\varepsilon }}}r_0 \lambda _1)}{\lambda _1 (1+ 6{{\mathrm{\varepsilon }}}r_0^2 + r_0^4) }, \quad \lambda _1' := ~ (\lambda _1 R_1)^{-1}, \quad a_{1 1}' := \frac{ 1 - 4 {{\mathrm{\varepsilon }}}r_0^2 - r_0^4 }{R_1 ({{\mathrm{\varepsilon }}}+ r_0^2 )^2}, \\ a_{2 1}' \!:= \!&-\frac{4 r_0}{R_1 ({{\mathrm{\varepsilon }}}+ r_0^2 )^2}, \quad \! \lambda _1 \!:=\! \frac{1 + 6 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4}{2 \sqrt{1 - 2 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4 }}, \quad \! c_1 \!:=\! ~ \frac{{{\mathrm{\mathrm {i}}}}r_0 \lambda _1 (-1+ 4 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4)}{({{\mathrm{\varepsilon }}}- r_0^2) (1 + 6 {{\mathrm{\varepsilon }}}r_0^2 + r_0^4)}, \\ u_1 :=&-1, \qquad u'_1 := -1, \qquad r_1' := 0, \qquad r_1 := 0. \end{aligned}$$

We give the standard parameters for \(\widetilde{H}\) for renormalizing \(\mathcal {H}_{2}^{-}\) in Lemma 6.2:

$$\begin{aligned} R_2 := ~&\left( \frac{1 + \sqrt{2} r_0 (e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + e^{{{\mathrm{\mathrm {i}}}}\theta _0})}{(1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0}) (1 + \sqrt{2} r_0 e^{{{\mathrm{\mathrm {i}}}}\theta _0})} \right) ^{1/2},\\ R_2' := ~&\frac{(1 + \sqrt{2} (e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + e^{{{\mathrm{\mathrm {i}}}}\theta _0}) r_0 + 2 r_0^2)^2 (2 (e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + e^{{{\mathrm{\mathrm {i}}}}\theta _0}) r_0 + \sqrt{2} (1 + 2 r_0^2))^2}{(1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0})^4 (1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^4 (1 + \sqrt{2} (e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + e^{{{\mathrm{\mathrm {i}}}}\theta _0}) r_0)^2} \\ c'_{1 2} := ~&\bigl ((e^{{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} r_0) (-c_2 u_2 (1 + 3 r_0^2 + 2 e^{2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0^2 + 2 \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 (1 + r_0^2) - {{\mathrm{\mathrm {i}}}}v_0) \\&+ {{\mathrm{\mathrm {i}}}}e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 (1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0) \lambda _2)\bigr ) \slash \bigl ((1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0) (e^{{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} r_0 + \sqrt{2} e^{2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0) \lambda _2\bigr ) \\ c'_{2 2} := ~&\bigl ((e^{{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} r_0) (c_2 u_2 (-r_0 (3 r_0 + 2 e^{2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0 + 2 \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} (1 + r_0^2)) + {{\mathrm{\mathrm {i}}}}v_0) \\&+ {{\mathrm{\mathrm {i}}}}e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 (1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0) \lambda _2)\bigr ) \slash \bigl ((1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0) (e^{{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} r_0 + \sqrt{2} e^{2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0) \lambda _2\bigr )\\ \lambda '_2 := ~&(\lambda _2 R_2)^{-1}\\ a'_{1 2} := ~&\frac{1 + 3 r_0^2 + 2 e^{-2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0^2 + 2 \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} r_0 (1 + r_0^2) + {{\mathrm{\mathrm {i}}}}v_0}{u_2 u_2' R_ 2 (1 + \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^2}\\ a'_{2 2} := ~&-\frac{3 r_0^2 + 2 e^{-2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0^2 + 2 \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} r_0 (1 + r_0^2) + {{\mathrm{\mathrm {i}}}}v_0}{u_2 u_2' R_2 (1 + \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^2} \\ u'_2 := ~&\frac{e^{{{\mathrm{\mathrm {i}}}}\theta _0} (\sqrt{2} r_0 \!+\! \sqrt{2} e^{-2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0 \!+\! e^{-{{\mathrm{\mathrm {i}}}}\theta _0} (1 \!+\! 2 r_0^2)) (2 r_0 + 2 e^{-2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0 + \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} (1 + 2 r_0^2))}{(1 + \sqrt{2} e^{-{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^4 (e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} r_0 + \sqrt{2} e^{-2 {{\mathrm{\mathrm {i}}}}\theta _0} r_0) R'_2 u_2^3} \\ u_2 := ~&\frac{2 R'_2 (1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0})^4 (1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^4 (1 + \sqrt{2} r_0e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0)}{(1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 + 2 r_0^2) (\sqrt{2} + 2 r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + 2 e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 + 2 r_0^2)^3}\\ \lambda _2 := ~&\frac{\sqrt{2} R_2' (1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0})^4 (1 + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^4 (1 + \sqrt{2} r_0e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0)^2}{(1 + \sqrt{2} r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + \sqrt{2} e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 + 2 r_0^2)^2 (\sqrt{2} + 2 r_0 e^{-{{\mathrm{\mathrm {i}}}}\theta _0} + 2 e^{{{\mathrm{\mathrm {i}}}}\theta _0} r_0 + 2 r_0^2)^2}. \end{aligned}$$

The remaining parameters \(c_2,r_2\) and \(r_2'\) are set to \(0\).

We give the standard parameters for the map \(H_3\) for the renormalization of \(\mathcal {H}_{3}^{-}\):

$$\begin{aligned} R_3 := ~&\sqrt{ \frac{-1 + r_0^4}{r_0^4}}, \qquad c'_{1 3} := ~ \frac{c_3 r_0^4}{\lambda _3 (1 - r_0^4)}, \qquad c'_{2 3} := \frac{r_0({{\mathrm{\mathrm {i}}}}c_3 r_0 + \lambda _3)}{\lambda _3 (1-r_0^4)}, \\ \lambda _3' :=&\, \Bigl (\lambda _3 R_3\Bigr )^{-1}, \quad a'_{1 3} := -{{\mathrm{\mathrm {i}}}}/R_3, \qquad a'_{2 3}:= 1/(r_0^2 R_3), \\ c_3 :=&\, \frac{{{\mathrm{\mathrm {i}}}}(-1 + 3 r_0^4)}{8 r_0^2}, \qquad \lambda _3 := \frac{-1 + r_0^4}{2 r_0},\qquad u'_3 := ~{{\mathrm{\mathrm {i}}}}, \end{aligned}$$

and the remaining parameters \(u_3, r_3'\) and \(r_3\) to be trivial.

If we consider \(H_4\) we renormalize the mapping \(\mathcal {H}_{4}^{{{\mathrm{\varepsilon }}}}=(f_{1 p_0},f_{2 p_0},g_{p_0})\). Here we use the following standard parameters, which only cover the case when \(g_{p_0 w}(0)>0\). If \(g_{p_0 w}(0)<0\) we need to interchange some of the standard parameters given here as described in the proof of Theorem 1.5:

$$\begin{aligned} R_4 := ~&\sqrt{3}\sqrt{\frac{{{\mathrm{\varepsilon }}}+ 14 r_0^4 + {{\mathrm{\varepsilon }}}r_0^8}{1+ 3 {{\mathrm{\varepsilon }}}r_0^4}}, \qquad c'_{1 4} := ~ \frac{4 c_4 r_0^2 u (-1 + r_0^4 {{\mathrm{\varepsilon }}}) - 8 {{\mathrm{\mathrm {i}}}}r_0^5 {{\mathrm{\varepsilon }}}\lambda _4}{(14 r_0^4 + {{\mathrm{\varepsilon }}}+ r_0^8 {{\mathrm{\varepsilon }}}) \lambda _4}, \\ c'_{2 4} := ~&\frac{c_4 u_4 (-1 + 3 r_0^8 + 14 r_0^4 {{\mathrm{\varepsilon }}}) - 8 {{\mathrm{\mathrm {i}}}}r_0^3 {{\mathrm{\varepsilon }}}\lambda _4}{\sqrt{3} (14 r_0^4 + {{\mathrm{\varepsilon }}}+ r_0^8 {{\mathrm{\varepsilon }}}) \lambda _4}, \qquad \lambda _4' := \Bigl (\lambda _4 R_4\Bigr )^{-1} \\ a'_{1 4} := ~&\frac{-12 r_0^2 (-1 + r_0^4 {{\mathrm{\varepsilon }}})}{u_4 u_4' R_4 (1 + 3 r_0^4 {{\mathrm{\varepsilon }}})^2}, \qquad \qquad \qquad \quad a'_{2 4}:= -\sqrt{3} \frac{1 - 3 r_0^8 - 14 r_0^4 {{\mathrm{\varepsilon }}}}{u_4 u_4' R_4 (1 + 3 r_0^4 {{\mathrm{\varepsilon }}})^2} \\ c_4 := ~&\frac{{{\mathrm{\mathrm {i}}}}r_0^3 (-7 - 26 r_0^8 + 9 r_0^{16} - 36 r_0^4 {{\mathrm{\varepsilon }}}+ 60 r_0^{12} {{\mathrm{\varepsilon }}}) \lambda _4}{u_4 (-19 r_0^4 - 38 r_0^{12} + 9 r_0^{20} - (1 + 74 r_0^8 - 123 r_0^{16}) {{\mathrm{\varepsilon }}})}\\ \lambda _4 := ~&\left( 4 \sqrt{3} r_0 \left| \frac{{{\mathrm{\varepsilon }}}- r_0^4}{1+ 14 {{\mathrm{\varepsilon }}}r_0^4 + r_0^8}\right| \right) ^{-1}, \qquad \quad u'_4 := \frac{{{\mathrm{sgn}}}(r_0^4 - {{\mathrm{\varepsilon }}})}{u_4^3 {{\mathrm{sgn}}}(1 + r_0^8 + 14 r_0^4 {{\mathrm{\varepsilon }}})} \\ u_4 := ~&\left( \frac{1 - {{\mathrm{\varepsilon }}}}{2}\right) \left( \frac{{{\mathrm{sgn}}}(-1 - 33 r_0^4 + 33 r_0^8 + r_0^{12})}{ {{\mathrm{sgn}}}(1 - 14 r_0^4 + r_0^8)}\right) \\ -&\, \left( \frac{1 + {{\mathrm{\varepsilon }}}}{2}\right) {{\mathrm{sgn}}}(-1 + 34 r_0^4 - 34 r_0^{12} + r_0^{16}). \end{aligned}$$

The remaining parameters \(r_4\) and \(r_4{'}\) are taken to be \(0\).

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Reiter, M. Classification of Holomorphic Mappings of Hyperquadrics from \(\mathbb {C}^2\) to \(\mathbb {C}^3\) . J Geom Anal 26, 1370–1414 (2016). https://doi.org/10.1007/s12220-015-9594-6

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