Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Every real ellipsoid in $ \mathbb{C}^2$ admits CR umbilical points

Authors: Xiaojun Huang and Shanyu Ji
Journal: Trans. Amer. Math. Soc. 359 (2007), 1191-1204
MSC (2000): Primary 32V40
Published electronically: August 15, 2006
MathSciNet review: 2262847
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every real ellipsoid $ M\subset\mathbb{C}^2$ admits at least four umbilical points, which can be compared to the result of Webster that a generic real ellipsoid in $ \mathbb{C}^n$ with $ n\ge3$ does not admit any umbilical point.

References [Enhancements On Off] (What's this?)

  • [BER] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings, Princeton Math. Ser. 47, Princeton Univ. Press, Princeton, NJ, 1999. MR 1668103 (2000b:32066)
  • [Car] É. Cartan, Sur les variétés pseudo-conformal des hypersurfaces de l'espace de deux variables complexes, Ann. Mat. Pura Appl. (4) 11, 17-90(1932).
  • [C] S. S. Chern, On the projective structure of a real hypersurface in $ C_{n+1}$, Math. Scand. 36, 74-82(1975). MR 0379910 (52:814)
  • [CJ1] S. S. Chern and S. Ji, Projective geometry and Riemann's mapping problem, Math. Ann. 302, 581-600(1995). MR 1339928 (96j:32027)
  • [CJ2] S. S. Chern and S. Ji, On the Riemann mapping theorem, Ann. of Math. 144, 421-439(1996). MR 1418903 (97m:32037)
  • [CM] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133, 219-271(1974). MR 0425155 (54:13112)
  • [F] J. Faran, Segre families and real hypersurfaces, Invent. Math. 60, 135-172(1980). MR 0586425 (81i:32018)
  • [Ham] H. Hamburger, Beweis einer Caratheodoryschen vermutung, Teil I. Ann. of Math. 41 (1940) 63-86; II. Acta Math. 73 (1941) 175-228; III. Acta Math 73 (1941) 229-332. MR 0001052 (1:172b); MR 0006480 (3:310a); MR 0006481 (3:310b)
  • [Hu] X. Huang, Local Equivalence Problems for Real Submanifolds in Complex Spaces, Lecture Notes in Mathematics 1848 (C.I.M.E. series), Springer-Verlag, pp. 109-161, Berlin-Heidelberg-New York, 2004. MR 2087582 (2005j:32041)
  • [HJ] X. Huang and S. Ji, Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism group of algebraic domains, Ann. Inst. Fourier (52), 1793-1831, 2002. MR 1954325 (2004c:32075)
  • [HJY] X. Huang, S. Ji and S.S.T. Yau, An example of a real analytic strongly pesudoconvex hypersurface which is not holomorphically equivalent to any algebraic hypersurface, Ark. Mat. (39), 75-93, 2001. MR 1821083 (2001m:32070)
  • [Jac] H. Jacobowitz, An introduction to CR geometry, Math Surveys and Monographs, AMS, 1990. MR 1067341 (93h:32023)
  • [Spv] M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume (3), Publish or Perish, Inc., Wilmington, Delaware, 1979. MR 0532832 (82g:53003c)
  • [We1] S. M. Webster , On the mapping problem for algebraic real hypersurfaces, Invention Math. 43, 53-68 (1977). MR 0463482 (57:3431)
  • [We2] S. M. Webster, Holomorphic differential invariants for an ellipsoidal real hypersurface, Duke Math J., 104(3), 463-475 (2000). MR 1781479 (2001i:32056)
  • [We3] S. M. Webster , A remark on the Chern-Moser tensor, Houston J. of Math., Vol. 28, No. 2, pp. 433-435 (2002). MR 1898199 (2003d:32047)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32V40

Retrieve articles in all journals with MSC (2000): 32V40

Additional Information

Xiaojun Huang
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Shanyu Ji
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204

Received by editor(s): December 9, 2004
Published electronically: August 15, 2006
Additional Notes: The first author was supported in part by NSF-0500626
Dedicated: To the memory of Professor S. S. Chern
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society