On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds
英文摘要：
Applying Lie's theory, we show that any $\mathcal{C}^ω$ hypersurface $M^5 \subset \mathbb{C}^3$ in the class $\mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at any point $p \in M$ to be the $v$-axis in coordinates $(z, ζ, w = u + i\, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $γ$ is the $v$-axis so that $M = \{u=F(z,ζ,\overline{z},\overlineζ,v)\}$ has Poincaré-Moser reduced equation: \begin{align} u & = z\overline{z} + \tfrac{1}{2}\,\overline{z}^2ζ+ \tfrac{1}{2}\,z^2\overlineζ + z\overline{z}ζ\overlineζ + \tfrac{1}{2}\,\overline{z}^2ζζ\overlineζ + \tfrac{1}{2}\,z^2\overlineζζ\overlineζ + z\overline{z}ζ\overlineζζ\overlineζ \\ & + 2{\rm Re} \{ z^3\overlineζ^2 F_{3,0,0,2}(v) + ζ\overlineζ ( 3\,{z}^2\overline{z}\overlineζ F_{3,0,0,2}(v) ) \} \\ & + 2{\rm Re} \{ z^5\overlineζ F_{5,0,0,1}(v) + z^4\overlineζ^2 F_{4,0,0,2}(v) + z^3\overline{z}^2\overlineζ F_{3,0,2,1}(v) + z^3\overline{z}\overlineζ^2 F_{3,0,1,2}(v) + z^3{\overlineζ}^3 F_{3,0,0,3}(v) \} \\ & + z^3\overline{z}^3 {\rm O}_{z,\overline{z}}(1) + 2{\rm Re} ( \overline{z}^3ζ{\rm O}_{z,ζ,\overline{z}}(3) ) + ζ\overlineζ\, {\rm O}_{z,ζ,\overline{z},\overlineζ}(5). \end{align} The values at the origin of Pocchiola's two primary invariants are: \[ W_0 = 4\overline{F_{3,0,0,2}(0)}, \quad\quad J_0 = 20\, F_{5,0,0,1}(0). \] The proofs are detailed, accessible to non-experts. The computer-generated aspects (upcoming) have been reduced to a minimum.
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