Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on CR mappings of positive codimension


Author: Jean-Charles Sunyé
Journal: Proc. Amer. Math. Soc. 138 (2010), 605-614
MSC (2000): Primary 32V10, 32H02, 32V20
DOI: https://doi.org/10.1090/S0002-9939-09-10062-X
Published electronically: September 25, 2009
MathSciNet review: 2557177
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following Artin type approximation theorem for smooth CR mappings: given $ M\subset \mathbb{C}^N$ a connected real-analytic CR submanifold that is minimal at some point, $ M^\prime\subset \mathbb{C}^{N^\prime}$ a real-analytic subset, and $ H\colon M\to M^\prime$ a $ \mathcal{C}^{\infty}$-smooth CR mapping, there exists a dense open subset $ \mathcal{O}\subset M$ such that for any $ q\in \mathcal{O}$ and any positive integer $ k$ there exists a germ at $ q$ of a real-analytic CR mapping $ H^k\colon (M,q)\to M^\prime$ whose $ k$-jet at $ q$ agrees with that of $ H$ up to order $ k$.


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

  • 1. M. Artin, On the solutions of analytic equations. Invent. Math. 5 (1968), 277-291. MR 0232018 (38:344)
  • 2. M. S. Baouendi, P. Ebenfelt, L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings. Princeton Math. Series 47, Princeton Univ. Press, 1999. MR 1668103 (2000b:32066)
  • 3. M. S. Baouendi, N. Mir, L. P. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space. J. Geom. Anal. 12 (2002), no. 4, 543-580. MR 1916859 (2003m:32035)
  • 4. M. S. Baouendi, H. Jacobowitz, F. Trèves, On the analyticity of CR mappings. Ann. of Math. (2) 122 (1985), no. 2, 365-400. MR 808223 (87f:32044)
  • 5. M. S. Baouendi, L. P. Rothschild, D. Zaitsev, Equivalences of real submanifolds in complex space. J. Differential Geom. 59 (2001), no. 2, 301-351. MR 1908985 (2003k:32051)
  • 6. S. Damour, Feuilletages holomorphes locaux et analyticité partielle d'applications CR $ \mathcal{C}^\infty$. Manuscripta Math. 109 (2002), no. 2, 203-222. MR 1935029 (2003h:32048)
  • 7. X. Huang, Local equivalence problems for real submanifolds in complex spaces. Real Methods in Complex and CR Geometry, Lecture Notes in Math. 1848, Springer, Berlin, 2004, 109-163. MR 2087582 (2005j:32041)
  • 8. X. Huang, W. Yin, A Bishop surface with a vanishing Bishop invariant. Invent. Math. 176 (2009), no 3, 461-520. MR 2501295
  • 9. F. Meylan, N. Mir, D. Zaitsev, Analytic regularity of CR-mappings. Math. Res. Lett. 9 (2002), no. 1, 73-93. MR 1892315 (2003d:32041)
  • 10. F. Meylan, N. Mir, D. Zaitsev, Approximation and convergence of formal -mappings. Int. Math. Res. Not. (2003), no. 4, 211-242. MR 1935273 (2004b:32058)
  • 11. F. Meylan, N. Mir, D. Zaitsev, On some rigidity properties of mappings between CR-submanifolds in complex space. Journées ``Équations aux Dérivées Partielles'', Exp. No. XII, 20 pp., Univ. Nantes, Nantes, 2003. MR 2050598 (2005a:32040)
  • 12. J. K. Moser, S. M. Webster, Normal forms for real surfaces in $ {C}\sp{2}$ near complex tangents and hyperbolic surface transformations. Acta Math. 150 (1983), no. 3-4, 255-296. MR 709143 (85c:32034)
  • 13. J. C. Sunyé, On formal maps between generic submanifolds in complex space. J. Geom. Anal. 19 (2009), no. 4, 944-962.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32V10, 32H02, 32V20

Retrieve articles in all journals with MSC (2000): 32V10, 32H02, 32V20


Additional Information

Jean-Charles Sunyé
Affiliation: Laboratoire de Mathématiques Raphaël Salem, Centre National de la Recherche Scientifique, Université de Rouen, Avenue de l’Université, B.P. 12, 76801 Saint Etienne du Rouvray, France
Email: jean-charles.sunye@etu.univ-rouen.fr

DOI: https://doi.org/10.1090/S0002-9939-09-10062-X
Keywords: CR mapping, Artin approximation theorem
Received by editor(s): March 16, 2009
Published electronically: September 25, 2009
Additional Notes: The author was partially supported by the Amadeus program of the “Partenariat Hubert Curien”.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society