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

## 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

Received: 3 March 2020; Revised: 23 August 2020; Accepted: 15 September 2020; Published: April, 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.11650/tjm/200903

Subjects:
Primary: 32V35, 32V40, 53-08, 53A55, 53BXX, 58K50
Secondary: 22E05, 22E60, 32A05, 53A07, 53B25, 58A15, 58A30

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 2 • April, 2021