2015/10/18 by Peter Ebenfelt, Ebenfelt, Peter, Bernhard Lamel +3
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.CV #msc:32H02 #msc:32V40
paper · pdf · doi:10.48550/arxiv.1510.05335
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arxiv created 2015/10/20 · arxiv updated 2015/10/21
In this paper, we study the real hypersurfaces M in \mathbb 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 an 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× \mathbb 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.