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
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