International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 5, Pages 281-302

Convergence of formal invertible CR mappings between minimal holomorphically nondegenerate real analytic hypersurfaces

Joël Merker

Laboratoire D'Analyse, Topologie et Probabilités, Centre de Mathématiques et D'Informatique, UMR 6632, 39 rue Joliot Curie, Marseille Cedex 13 F-13453, France

Received 24 October 2000; Revised 19 April 2001

Copyright © 2001 Joël Merker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Recent advances in CR (Cauchy-Riemann) geometry have raised interesting fine questions about the regularity of CR mappings between real analytic hypersurfaces. In analogy with the known optimal results about the algebraicity of holomorphic mappings between real algebraic sets, some statements about the optimal regularity of formal CR mappings between real analytic CR manifolds can be naturally conjectured. Concentrating on the hypersurface case, we show in this paper that a formal invertible CR mapping between two minimal holomorphically nondegenerate real analytic hypersurfaces in n is convergent. The necessity of holomorphic nondegeneracy was known previously. Our technique is an adaptation of the inductional study of the jets of formal CR maps which was discovered by Baouendi-Ebenfelt-Rothschild. However, as the manifolds we consider are far from being finitely nondegenerate, we must consider some new conjugate reflection identities which appear to be crucial in the proof. The higher codimensional case will be studied in a forthcoming paper.