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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
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Taiwanese J. Math. 25(2): 333-364 (April, 2021). DOI: 10.11650/tjm/200903


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.


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


Download 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.


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

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


Vol.25 • No. 2 • April, 2021
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