Geodesic
Normal Form for Infinite Type Hypersurfaces in $\bbb C^2$ with Nonvanishing Levi Form Derivative
Documenta mathematica, Tome 22 (2017), pp. 165-190.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

In this paper, we study real hypersurfaces $M$ in $\Bbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of a universal polynomial in the 7-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely the Chern-Moser normal form at Levi-nondegenerate points. For a fixed hypersurface, its normal forms are parametrized by $S^1\times \Bbb R^\ast$, and as a corollary we find that the automorphisms in the stability group of $M$ at $p$ without resonances are determined by their 1-jets at $p$. In the last section, as a contrast, we also give examples of hypersurfaces with arbitrarily high resonances that possess families of distinct automorphisms whose jets agree up to the resonant order.
Classification : 32V40
Keywords: real hypersurfaces in $\Bbb C^2$, points of infinite type, normal forms 
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     author = {Ebenfelt, P. and Lamel, B. and Zaitsev, D.},
     title = {Normal {Form} for {Infinite} {Type} {Hypersurfaces} in $\bbb C^2$ with {Nonvanishing} {Levi} {Form} {Derivative}},
     journal = {Documenta mathematica},
     pages = {165--190},
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     volume = {22},
     year = {2017},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a46/}
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Ebenfelt, P.; Lamel, B.; Zaitsev, D. Normal Form for Infinite Type Hypersurfaces in $\bbb C^2$ with Nonvanishing Levi Form Derivative. Documenta mathematica, Tome 22 (2017), pp. 165-190. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a46/