Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $df$) are constant $\sigma_1,\sigma_2$.
Question: Do there exist smooth isometries $ \phi_1, \phi_2:U \to U$ such that $\phi_1 \circ f \circ \phi_2=f^{-1}$? ($\phi_i $ must be affine; I want them to map $U$ into $U$.)
Note that $df^{-1}=(df)^{-1}$ has singular values $\sigma_1,\sigma_2$ the same as $df$.
Here are two examples for this phenomena:
1. Affine maps on ellipses:
Let $0<a<b$, $ab=1$, and let $$ U=U_{a,b}=\biggl\{(x,y) \,\biggm | \, \frac{x^2}{a^2} + \frac{y^2}{b^2} < 1 \biggr\}. $$
Take $f(x,y)=A\pmatrix{x\\y}$, where $$\begin{align*} & A=A(\theta)= \begin{pmatrix} a& 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos \theta \end{pmatrix}\begin{pmatrix} 1/a& 0 \\ 0 & 1/b \end{pmatrix}= \begin{pmatrix} \cos\theta & -\frac ab \sin\theta \\ \frac ba \sin\theta & \cos \theta \end{pmatrix} \end{align*}.$$
Then $A(\theta)^{-1}=A(-\theta)=JA(\theta) J$, where $J=\begin{pmatrix} 1& 0 \\ 0 & -1 \end{pmatrix}$ is the reflection around the $y$ axis.
2. Non-affine maps on the disk:
Let $U=D\setminus\{0\}$ where $D \subseteq \mathbb R^2$ is the unit disk.
$f_c: (r,\theta)\to (r,\theta+c\log r )$. Then we have $f_{c}^{-1}=f_{-c}=Jf_{c}J$.
Note that $ [df_c]_{\{ \frac{\partial}{\partial r},\frac{1}{r}\frac{\partial}{\partial \theta}\}}=\begin{pmatrix} 1 & 0 \\\ c & 1\end{pmatrix}, $ so the singular values of $f_c$ are constants which depend on $c$.
There are many local solutions to the PDE $\sigma_i(df)=\sigma_i$, so I don't expect $f$ and $f^{-1}$ to be the same up to isometries in general.
