Local Conformal Flatness of Left-Invariant 3D Contact Structures

In this talk I want to address the problem of finding the locally flat left-invariant contact structures on a three dimensional Lie Group up to conformal transformations, that is I will determine the ones locally conformally equivalent to the Heisenberg algebra $\mathbb H_3$. In particular I will show how to build the Fefferman metric associated to a generic three dimensional contact structure (not necessarily left-invariant) and by means of this construction I will give the explicit formula for the (unique) conformal invariant associated to such a structure. Next, specializing the study to the left-invariant case, I will give a complete list of the locally conformally flat structures which may appear and I will find the explicit form of the maps $\varphi: M\to \mathbb R$ which flatten our structures, and I will show that they are essentially (i.e. up to multiplication by a constant) unique.
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Theorem. Let $(M,\Delta,g)$ be a left-invariant 3D contact structure. Then it is locally conformally flat if and only if its canonical frame satisfies one of the following
\begin{equation*} \text{i)}\; \left\{ \begin{array}{lll} [f_2,f_1]&=&f_0+c_{12}^2f_2,\\
\\
[f_1,f_0]&=&\frac29\left(c_{12}^2\right)^2f_2,\\
\\
[f_2,f_0]&=&0.
\end{array} \right.\qquad\text{ii)}\; \left\{ \begin{array}{lll} [f_2,f_1]&=&f_0+c_{12}^1f_1,\\
\\
[f_1,f_2]&=&0,\\
\\
[f_2,f_0]&=&-\frac29\left(c_{12}^1\right)^2f_2.
\end{array} \right. \end{equation*}
or
\begin{equation*} \text{iii)}\; \left\{ \begin{array}{lll} [f_2,f_1]&=&f_0,\\
[f_1,f_0]&=&\kappa f_2,\\
[f_2,f_0]&=&-\kappa f_1,\qquad \kappa<0. \end{array} \right. \end{equation*}
Where $\kappa$ is the curvature of the structure.
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Open question 1. Give a complete classification (i.e. not just the locally conformally flat ones) of left-invariant three dimesional contact structures, up to real rescalings.
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Open question 2. Give satisfactory criteria to determine whether a given three dimesional contact structure (not necessarily left-invariant) is locally conformally flat or not.