Normal Form for Infinite Type Hypersurfaces in C2\mathbb{C}^2 with Nonvanishing Levi Form Derivative

  • P. Ebenfelt

    Department of Mathematics University, of California at San Diego, La Jolla, CA 92093-0112
  • B. Lamel

    Department of Mathematics, University of Vienna, Vienna, Austria
  • D. Zaitsev

    School of Mathematics, Trinity College Dublin, Dublin 2, Ireland
Normal Form for Infinite Type Hypersurfaces in $\mathbb{C}^2$ with Nonvanishing Levi Form Derivative cover
Download PDF

This article is published open access.

Abstract

In this paper, we study real hypersurfaces MM in C2\Bbb C^2 at points pMp\in M of infinite type. The degeneracy of MM at pp 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 MM. The main result is a complete (formal) normal form at points pp 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×RS^1\times \Bbb R^\ast, and as a corollary we find that the automorphisms in the stability group of MM at pp without resonances are determined by their 1-jets at pp. 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.

Cite this article

P. Ebenfelt, B. Lamel, D. Zaitsev, Normal Form for Infinite Type Hypersurfaces in C2\mathbb{C}^2 with Nonvanishing Levi Form Derivative. Doc. Math. 22 (2017), pp. 165–190

DOI 10.4171/DM/563