Normal Forms for Rigid ${\mathfrak{C}}_{2,1}$ Hypersurfaces $M^5\subset {\mathbb{C}}^3$

Abstract

Consider a $2$-nondegenerate constant Levi rank $1$ rigid ${\mathcal{C}}^{\omega }$ hypersurface $M^5\subset {\mathbb{C}}^3$ in coordinates $(z,\zeta ,w=u+iv)$: \[ u = F\big(z,ζ,\bar{z},\barζ\big). \] The Gaussier-Merker model $u=\frac{z\bar{z}+\frac{1}{2}{z}^2\bar{\zeta }+\frac{1}{2}{\bar{z}}^2\zeta }{1-\zeta \bar{\zeta }}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone $\{x_1^2+x_2^2-x_3^2=0\}$. Another representation is the tube $u=\frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: \[ (z,ζ,w) \longmapsto \big( f(z,ζ), g(z,ζ), ρ\,w+h(z,ζ) \big) =: (z',ζ',w'). \] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiola's formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\bar{z}+\frac{1}{2}\,z^2\barζ +\frac{1}{2}\,\bar{z}^2ζ}{ 1-ζ\barζ} + \sum_{a,b,c,d \atop a+c\geqslant 3}\, G_{a,b,c,d}\, z^aζ^b\bar{z}^c\barζ^d, \] with $0=G_{a,b,0,0}=G_{a,b,1,0}=G_{a,b,2,0}$ and $0=G_{3,0,0,1}=\mathrm{Im}{G}_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0,2,3,0}$, $\mathrm{Re}{G}_{3,0,1,1}$. With this, a brige Poincaré $⟷$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.

This work was supported in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583

References in corpus (1)

• I: Lie symmetries of partial differential equations and CR geometry