# Normal Form for Infinite Type Hypersurfaces in $\mathbb{C}^2$ 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

## Abstract

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

## Cite this article

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

DOI 10.4171/DM/563