Normal forms and biholomorphic equivalence of real hypersurfaces in C^3
Abstract
We consider the problem of describing the local biholomorphic equivalence class of a realanalytic hypersurface $M$ at a distinguished point $p_0\in M$ by giving a normal form for such objects. In order for the normal form to carry useful information about the biholomorphic equivalence class, we shall require that the transformation to normal form is unique modulo some finite dimensional group. A classical result due to ChernMoser gives such a normal form for Levi nondegenerate hypersurfaces. The main results in this paper concern realanalytic hypersurfaces $M$ in $\Bbb C^3$ at certain Levi degenerate points $p_0\in M$, namely points at which $M$ is 2nondegenerate. We give a partial normal form for all such $(M,p_0)$, i.e. a normal form for the data associated with 2nondegeneracy. We also give a complete formal normal form for such $(M,p_0)$ under the additional condition that the Levi form has rank one at $p_0$. This result, combined with a recent theorem due to the author, M. S. Baouendi, and L. P. Rothschild stating that formal equivalences between realanalytic finitely nondegenerate hypersurfaces converge, gives a description of the biholomorphic equivalence class of a realanalytic hypersurface in $\Bbb C^3$ at a point of 2nondegeneracy where the rank of the Levi form is one.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1997
 DOI:
 10.48550/arXiv.math/9705201
 arXiv:
 arXiv:math/9705201
 Bibcode:
 1997math......5201E
 Keywords:

 Mathematics  Complex Variables;
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