On the CR Transversality of Holomorphic Maps into Hyperquadrics

Dedicated to Professor Yum-Tong Siu on the occasion of his 70th birthday
  • Xiaojun HuangEmail author
  • Yuan Zhang
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)


Let M be a smooth Levi-nondegenerate hypersurface of signature in C n with n ≥ 3, and write H N for the standard hyperquadric of the same signature in C N with \(N - n <\frac{n-1} {2}\). Let F be a holomorphic map sending M into \(H_{\ell}^{N}\). Assume F does not send a neighborhood of M in C n into \(H_{\ell}^{N}\). We show that F is necessarily CR transversal to M at any point. Equivalently, we show that F is a local CR embedding from M into \(H_{\ell}^{N}\).


Power Series Expansion Open Dense Subset Quantitative Version Weighted Degree Rigidity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Part of work was done when the authors were attending the 7th Workshop on geometric analysis of PDE and several complex variables at Serra Negra, Brazil. Both authors would like to thank the organizers for the kind invitation. The second author is also indebted to Wuhan University for the hospitality during her several visits there.


  1. 1.
    Baouendi, M.S., Huang, X.: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Differ. Geom. 69, 379–398 (2005)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Baouendi, M.S., Rothschild, L.P.: Geometric properties of mappings between hypersurfaces in complex space. J. Differ. Geom. 31(2), 473–499 (1990)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Baouendi, M.S., Huang, X., Rothschild, L.P.: Nonvanishing of the differential of holomorphic mappings at boundary points. Math. Res. Lett. 2(6), 737–750 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematics Series, vol. 47. Princeton University Press, Princeton (1999)Google Scholar
  5. 5.
    Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Transversality of holomorphic mappings between real hypersurfaces in different dimensions. Comm. Anal. Geom. 15(3), 589–611 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics, pp. xiv+272. CRC Press, Boca Raton, FL (1993). ISBN: 0-8493-8272-6Google Scholar
  8. 8.
    Ebenfelt, P., Rothschild, L.R.: Transversality of CR mappings. Am. J. Math. 128, 1313–1343 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ebenfelt, P., Son, D.: Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions. Ill. J. Math. 56(1), 33–51 (2012/2013)Google Scholar
  10. 10.
    Ebenfelt, P., Huang, X., Zaitsev, D.: The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics. Am. J. Math. 127(1), 169–191 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fornaess, J.E.: Biholomorphic mappings between weakly pseudoconvex domains. Pac. J. Math. 74, 63–65 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang, X.: Schwarz reflection principle in complex spaces of dimension two. Comm. Partial Differ. Equ. 21(11–12), 1781–1828 (1996)zbMATHGoogle Scholar
  13. 13.
    Huang, X.: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. J. Differ. Geom. 51, 13–33 (1999)zbMATHGoogle Scholar
  14. 14.
    Huang, X., Zhang, Y.: Monotonicity for the Chern-Moser-Weyl curvature tensor and CR embeddings. Sci. China Ser. A 52(12), 2617–2627 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Huang, X., Zhang, Y.: On a CR transversality problem through the approach of the Chern-Moser theory. J. Geom. Anal. 23(4), 1780–1793 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Meylan, F., Mir, N., Zaitsev, D.: Approximation and convergence of formal CR-mappings. Int. Math. Res. Not. 2003(4), 211–242 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Mir, N.: Convergence of formal embeddings between real-analytic hypersurfaces in codimension one. J. Differ. Geom. 62(1), 163–173 (2002)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Pinchuk, S.: Proper holomorphic maps of strictly pseudoconvex domains. (Russian) Sibirsk. Mat. Z. 15, 909–917, 959 (1974)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Partially supported in part by National Science Foundation DMS-1363418, Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Partially supported in part by National Science Foundation DMS-1265330, Department of Mathematical SciencesIndiana University – Purdue UniversityFort WayneUSA

Personalised recommendations