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On Convergent Poincar'e-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds | | Wei Guo Foo
; Joel Merker
; The-Anh Ta
; | | Date: |
4 Mar 2020 | | Abstract: | Applying Lie’s theory, we show that any $mathcal{C}^omega$ 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 $(z, zeta, w = u + i, v)$ centered at $p$, we
show that there exists a (unique up to 5 parameters) convergent change of
complex coordinates fixing the origin in which $gamma$ is the $v$-axis so that
$M = {u=F(z,zeta,overline{z},overline{zeta},v)}$ has Poincar’e-Moser
reduced equation: egin{align} u & = zoverline{z} +
frac{1}{2},overline{z}^2zeta + frac{1}{2},z^2overline{zeta} +
zoverline{z}zetaoverline{zeta} +
frac{1}{2},overline{z}^2zetazetaoverline{zeta} +
frac{1}{2},z^2overline{zeta}zetaoverline{zeta} +
zoverline{z}zetaoverline{zeta}zetaoverline{zeta} \ & + 2{
m Re} {
z^3overline{zeta}^2 F_{3,0,0,2}(v) + zetaoverline{zeta} (
3,{z}^2overline{z}overline{zeta} F_{3,0,0,2}(v) ) } \ & + 2{
m Re} {
z^5overline{zeta} F_{5,0,0,1}(v) + z^4overline{zeta}^2 F_{4,0,0,2}(v) +
z^3overline{z}^2overline{zeta} F_{3,0,2,1}(v) +
z^3overline{z}overline{zeta}^2 F_{3,0,1,2}(v) + z^3{overline{zeta}}^3
F_{3,0,0,3}(v) } \ & + z^3overline{z}^3 {
m O}_{z,overline{z}}(1) + 2{
m
Re} ( overline{z}^3zeta {
m O}_{z,zeta,overline{z}}(3) ) +
zetaoverline{zeta}, {
m O}_{z,zeta,overline{z},overline{zeta}}(5).
end{align}
The values at the origin of Pocchiola’s two primary invariants are: [ W_0 =
4overline{F_{3,0,0,2}(0)}, quadquad 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. | | Source: | arXiv, 2003.1952 | | Services: | Forum | Review | PDF | Favorites | |
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