论文标题
在趋同的Poincaré-Moser减少李维归化嵌入$ 5 $二维的Cr歧管上
On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds
论文作者
论文摘要
应用Lie的理论，我们表明，任何$ \ MATHCAL {C}^ω$ hypersurface $ m^5 \ subset \ subset \ mathbb {c}^3 $ in Class $ \ Mathfrak {c} _ {c} _ {2,1} $携带cartan cartan cartan cartan-moser of Cartan-moser ofers cartan orders ofers cartan orders of cartan-moser $ 1 $ $ 1 $和$ 2 $。 集成并拉直$ 2 $ 2 $链条以$ p \ in m $中的$ v $ - 轴为coordinates $（z，ζ，ζ，w = u + i \，v，v，v）$中心的$ p $，我们表明存在一个（唯一的5个参数）固定$γ$ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v-max in \ {u = f（z，ζ，\ + overline {z}，\overlineζ，v）\} $具有poincaré-moser降低方程：\ begin {align} u＆= z \ z \ overline {z} {z} + \ tfrac {1} {1} {1} {1} {2} {2} {2} {Z} \ tfrac {1} {2} \，z^2 \overlineζ + z \ overline {z} prline {z}ζ\ overline队 + \ tfrac {1} {2} {2} \，\，\，\，\，\ edline {z} {z} {z}^2ζqumprac z \ ederline {z}ζ\overlineζζ\overlineζ\＆ + 2 {\ rm re} \ {z^3 \overlineζ^2 f_ {3,0,0,0,2}（v） +ζ +ζ\ everlinellineline（3 \，{z}^2 \，{z}^2 \ etlline {z}^z}} ）\} \\＆ + 2 {\ rm re} \ {z^5 \ overline面z^3 \ edline {z} \ overline面+ 2 {\ rm re}（\ edrowline {z}^3ζ{\ rm o} _ {z，ζ，\ overline {z}}}（3）） +ζ\序列\overlineζ\，{\ rm O} _ {z，z，z，em {z，q，q，quberline fline x，x，x z {z} { \ end {align} Pocchiola的两个主要不变性的起源值为：\ [W_0 = 4 \ overline {f_ {3,0,0,2}（0）}，\ Quad \ Quad \ Quad j_0 = 20 \，f_，f_，f_ {5,0,0,1}（0）。 \] 证明是详细的，可以通过非专家访问。计算机生成的方面（即将到来）已减少到最低。
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.
