Abstract
In this paper, we show that all CR immersions from smooth Levi-nondegenerate hypersurfaces into hyperquadrics with vanishing CR second fundamental forms are necessarily linear fractional.
Similar content being viewed by others
References
Alexander, H.: Holomorphic mappings from the ball and polydisc. Math. Ann. 209, 249–256 (1974)
Baouendi, M.S., Ebenfelt, P., Huang, X.: Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics. Adv. Math. 219(5), 1427–1445 (2008)
Baouendi, M.S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. Am. J. Math. 133(6), 1633–1661 (2011)
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)
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)
Baouendi, M.S., Huang, X.: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Differ. Geom. 69, 379–398 (2005)
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)
Baouendi, M.S., Rothschild, L.P.: Geometric properties of mappings between hypersurfaces in complex space. J. Differ. Geom. 31(2), 473–499 (1990)
Cheng, X., Ji, S.: Linearity and second fundamental forms for proper holomorphic maps from \({B}^{n+1}\) to \({B}^{4n-3}\). J. Geom. Anal. 22(4), 977–1006 (2012)
Cheng, X., Ji, S., Liu, W.: CR submanifolds in a sphere and their Gauss maps. Sci. China Math. 56(5), 1041–1049 (2013)
Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)
Cima, J., Suffridge, T.J.: A reflection principle with applications to proper holomorphic mappings. Math. Ann. 265, 489–500 (1983)
D’Angelo, J. P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1993)
Ebenfelt, P.: Partial rigidity of degenerate CR embeddings into spheres. Adv. Math. 239, 72–96 (2013)
Ebenfelt, P., Huang, X., Zaitsev, D.: Rigidity of CR-immersions into spheres. Commun. Anal. Geom. 12(3), 631–670 (2004)
Ebenfelt, P., Huang, X., Zaitsev, D.: The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics. Am. J. Math. 127(1), 169–191 (2005)
Ebenfelt, P., Shroff, R.: Partial rigidity of CR embeddings of real hypersurfaces into hyperquadrics with small signature difference. Commun. Anal. Geom. 23(1), 159–190 (2015)
Faran, J.J.: Maps from the two-ball to the three-ball. Invent. Math. 68(3), 441–475 (1982)
Forstneric, F.: Proper holomorphic maps from balls. Duke Math. J. 53, 427–441 (1986)
Hamada, H.: Rational proper holomorphic maps from \({\mathbb{B}}^n\) into \({\mathbb{B}}^{2n}\). Math. Ann. 331(3), 693–711 (2005)
Huang, X.: Schwarz reflection principle in complex spaces of dimension two. Commun. Partial Differ. Equ. 21(11–12), 1781–1828 (1996)
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)
Huang, X.: On a semi-rigidity property for holomorphic maps. Asian J. Math. 7(4), 463–492 (2003)
Huang, X., Ji, S.: Mapping \({\mathbb{B}^{n}}\) into \({\mathbb{B}^{2n-1}}\). Invent. Math. 145(2), 219–250 (2001)
Huang, X., Ji, S., Xu, D.: A new gap phenomenon for proper holomorphic mappings from \({\mathbb{B}^{n}}\) into \({\mathbb{B}^{n}}\). Math. Res. Lett. 13(4), 515–529 (2006)
Huang, X., Ji, S., Yin, W.: On the third gap for proper holomorphic maps between balls. Math. Ann. 358(1–2), 115–142 (2014)
Huang, X., Zhang, Y.: Monotonicity for the Chern–Moser–Weyl curvature tensor and CR embeddings. Sci. China Ser. A 52(12), 2617–2627 (2009). (Special issue in honor of Professor Tong-De Zhong)
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)
Huang, X., Zhang, Y.: On the CR transversality of holomorphic maps into hyperquadrics, to appear in Abel Symposia, dedicated to Professor Yum-Tong Siu on the occasion of his 70th birthday
Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems. Graduate Studies in Mathematics, vol. 61. American Mathematical Society, Providence (2003)
Ji, S., Yuan, Y.: Flatness of CR submanifolds in a sphere. Sci. China Math. 53(3), 701–718 (2010). (Special Issue Dedicated to Professor Yang Lo)
Lamel, B.: Holomorphic maps of real submanifolds in complex spaces of different dimensions. Pac. J. Math 201(2), 357–387 (2001)
Pinchuk, S.: Proper holomorphic maps of strictly pseudoconvex domains. Sibirsk. Mat. Z. 15, 909–917 (1974). (Russian)
Webster, S.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13(1), 25–41 (1978)
Acknowledgments
Both authors would like to thank Professor X. Huang and S. Ji for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Yuan Yuan: Supported in part by National Science Foundation Grant DMS-1412384. Yuan Zhang: Supported in part by National Science Foundation Grant DMS-1501024.
Rights and permissions
About this article
Cite this article
Yuan, Y., Zhang, Y. CR submanifolds with vanishing second fundamental forms. Geom Dedicata 183, 169–180 (2016). https://doi.org/10.1007/s10711-016-0152-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-016-0152-6