Recent advances in CR 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 $\C^n$ is convergent. The necessity of holomorphic nondegeneracy was known previously. Our technique is an adaptation 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 in studied in the two forthcoming papers http://fr.arxiv.org/abs/math.CV/0005290/v2 and http://fr.arxiv.org/abs/math.CV/0404248.
Publié le : 2001-07-05
Classification:
reflection function,
holomorphic nondegeneracy in the sense of N. Stanton,
hierarchy of nondegeneracy conditions,
conjugate reflection identities,
Formal CR mappings,
real analytic hypersurfaces,
32V25, 32V35, 32V40,
[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG],
[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV]
@article{hal-00003378,
author = {Merker, Joel},
title = {On the convergence of formal CR maps between hypersurfaces},
journal = {HAL},
volume = {2001},
number = {0},
year = {2001},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00003378}
}
Merker, Joel. On the convergence of formal CR maps between hypersurfaces. HAL, Tome 2001 (2001) no. 0, . http://gdmltest.u-ga.fr/item/hal-00003378/