Lie-Cartan differential invariants and Poincaré-Moser normal forms: Conflunces
by Zhangchi Chen   Wei Guo Foo   Joël Merker   The-Anh Ta  

Vol. 18 No. 2 (2023) P.133~P.184

DOI:	https://doi.org/10.21915/BIMAS.2023202
	10.21915/BIMAS.2023202

ABSTRACT

We study $2$-nondegenerate constant Levi rank $1$ rigid $\mathcal{C}^\omega$ hypersurfaces $M^5 \subset \mathbb{ C}^3$ with $0 \in M^5$ given in coordinates $(z, \zeta, w = u + iv)$ as $u = F\big(z,\zeta,\overline{z},\overline{\zeta}\big)$ under rigid biholomorphisms: \[ (z,\zeta,w) \,\,\,\longmapsto\,\,\, \big( f(z,\zeta),\, g(z,\zeta),\, \rho\,w+h(z,\zeta) \big) \,\,=:\,\, (z',\zeta',w'). \] In a previous article, a Cartan-type reduction to an $\{e\}$-structure was done by Foo-Merker-Ta. Three relative invariants appeared: V$_0$, I$_0$ (primary) and Q$_0$ (derived).

On the other hand, a Poincar$\acute{e}$-Moser complete normal form:

\[ u \,=\, \tfrac{z\overline{z}+\frac{1}{2}\,z^2\overline{\zeta} +\frac{1}{2}\,\overline{z}^2\zeta}{ 1-\zeta\overline{\zeta}} + \sum_{a,b,c,d\in\Bbb{N} \atop a+c\geqslant 3}\, G_{a,b,c,d} \big(F_{{\scriptscriptstyle{\bullet}}}\big)\, z^a\zeta^b\overline{z}^c\overline{\zeta}^d, \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {\rm Im}\, G_{1,1,3,0}$, has been recently obtained by the authors.

The model $u = \frac{z\overline{z}+\frac{1}{2}\,z^2\overline{\zeta} +\frac{1}{2}\,\overline{z}^2\zeta}{ 1-\zeta\overline{\zeta}}$ is equivalent to the future light cone $({\rm Im}\, z_0)^2 = ({\rm Im}\,z_1)^2 + ({\rm Im}\,z_2)^2$ with ${\rm Im}\,z_0 > 0$ deeply investigated by Sergeev.

In order to compare the two approaches, we compute (relative) invariants at every point, not only at the central point, and we 'discover' the proportionalities: \[ G_{0, 1, 4, 0} \big(F_{{\scriptscriptstyle{\bullet}}}\big) \,\propto\, V_0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ G_{0, 2, 3, 0} \big(F_{{\scriptscriptstyle{\bullet}}}\big) \,\propto\, I_0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm Re}\, G_{1,1,3,0} \big(F_{{\scriptscriptstyle{\bullet}}}\big) \,\propto\, Q_0. \]

With this, a bridge Poincar$\acute{e}$ $\longleftrightarrow$ Cartan is constructed.

In terms of $F$, the numerators of V$_0$, I$_0$, Q$_0$ incorporate $11$, $52$, $824$ differential monomials.

KEYWORDS
Normal forms, power series, explicit differential invariants, Levi degenerate CR manifolds, rigid CR manifolds

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 32V40, 58K50, 32V35, 53A55, 53-08. Secondary: 58A15, 32A05, 53A07, 53B25, 22E05, 22E60, 58A30

MILESTONES

Received: 2022-10-18
Revised :
Accepted:

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