The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$
How to compute all the local automorphisms of $\partial \Omega_{KN}$?
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What I have tried:
- I change the variable as the following $\widetilde{w}=iw,\ w=x+iy,\ x,y \in \Bbb R$. Then $\text{Re}\ \widetilde{w}=\text{Re}\ (-y+ix)=-y=-\text{Im}\ {w}$. We consider $r$ be a local defining function: $$\begin{align*} r:\Bbb C^2 & \to \Bbb R \\ (z,w)&\mapsto r(z,w)=\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6 \end{align*}$$
Thus, We have the equation of $\partial \Omega_{KN}=\{r=0\}$ in neighbourhood of $(0,0)$: \begin{align*} 0 &=\text{Re}\ \widetilde{w}+|iz\widetilde{w}|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6 \\ \Rightarrow \text{Im}w &= |z{w}|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6\\ \end{align*}
- By Martin Kolář, we have: $$P_{a}^{k,l}(z,\overline{z})=|z|^k+a|z|^{k-l}\text{Re}z^l$$
In case, we have $k=8,l=6,a=\dfrac{15}{7}$, and $\Omega_{KN}$ is a pseudoconvex.
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Now, I have stuck...when I try to compute all the local automorphisms! I think that we have $6$ local automorphisms. Because, we recall a Theorem:
The automorphism group of the Kohn - Nirenberg domain $\Omega_{KN}$ is equal to the set $\{\Pi^n:\ n=1, \ldots,6\}$, where $\Pi(z,w)=(e^\frac{i\pi}{3}z,w).$
But How to compute all the local automorphisms of $\partial \Omega_{KN}$?
Any help will be appreciated! Thanks!