Normal Form for Infinite Type Hypersurfaces in C^2 with Nonvanishing Levi Form Derivative

In this paper, we study real hypersurfaces $M$ in ${\ C}^{2}$ 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 $S^{1}\times {\
R}^*$, 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.

2010 Mathematics Subject Classification: 32H02, 32V40

Keywords and Phrases:

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