Geodesic
Normal Form for Infinite Type Hypersurfaces in $\mathbb{C}^2$ with Nonvanishing Levi Form Derivative
Documenta mathematica, Tome 22 (2017), pp. 165-190
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In this paper, we study real hypersurfaces M in C2 at points p∈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 S1×R∗, 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.
DOI : 10.4171/dm/563
Classification : 32V40
Mots-clés : normal forms, real hypersurfaces in C2, points of infinite type 
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     author = {P. Ebenfelt and B. Lamel and D. Zaitsev},
     title = {Normal {Form} for {Infinite} {Type} {Hypersurfaces} in $\mathbb{C}^2$ with {Nonvanishing} {Levi} {Form} {Derivative}},
     journal = {Documenta mathematica},
     pages = {165--190},
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P. Ebenfelt; B. Lamel; D. Zaitsev. Normal Form for Infinite Type Hypersurfaces in $\mathbb{C}^2$ with Nonvanishing Levi Form Derivative. Documenta mathematica, Tome 22 (2017), pp. 165-190. doi: 10.4171/dm/563

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