## Abstract

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathscr{C}^{\omega}$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, \zeta, w = u+iv)$: \[ u = F(z,\zeta,\overline{z},\overline{\zeta}). \] The Gaussier-Merker model $u = \frac{z \overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}}$ 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 $u = \frac{(\operatorname{Re}z)^2}{1 - \operatorname{Re} \zeta}$. The Gaussier-Merker model has $7$-dimensional rigid automorphisms group.

Inspired by Alexander Isaev, we study *rigid* biholomorphisms: \[ (z,\zeta,w) \longmapsto (f(z,\zeta), g(z,\zeta), \rho w + h(z,\zeta)) =: (z',\zeta',w'). \] The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}} + \sum_{\substack{a,b,c,d \in \mathbb{N} \\ a+c \geq 3}} G_{a,b,c,d} z^a \zeta^b \overline{z}^c \overline{\zeta}^d \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = \operatorname{Im} G_{3,0,1,1}$.

## Funding Statement

The realization of this research work in Cauchy-Riemann
(CR) geometry has received generous financial support from the scientific grant
2018/29/B/ST1/02583 originating from the Polish National Science Center (NCN). The second
author is supported by NSFC grant number 11688101.

## Acknowledgments

Grateful thanks are addressed to an anonymous referee for clever suggestions and a careful reading.

## Citation

Zhangchi Chen. Wei Guo Foo. Joël Merker. The Anh Ta. "Normal Forms for Rigid $\mathfrak{C}_{2,1}$ Hypersurfaces $M^5 \subset \mathbb{C}^3$." Taiwanese J. Math. 25 (2) 333 - 364, April, 2021. https://doi.org/10.11650/tjm/200903

## Information