Normal form for infinite type hypersurfaces in \(\mathbb C^2\) with nonvanishing Levi form derivative (Q510745)

Summary: In this paper, we study real hypersurfaces \(M\) in \(\mathbb C^2\) 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 \(S^1\times \mathbb R^\ast\), 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.