Since $\mathbb R^2$ is a non-hyperbolic surface, it is ignored in most accounts of Teichmuller theory. I look for papers where the space of conformal structures on $\mathbb R^2$, and the associated Teichmuller space is studied. What is known about the topological properties of these space?

EDIT: I looked up the definition of a conformal structure and realized that the above question is silly. Indeed, a conformal structure on an oriented $2$-manifold is a reduction of the structure group of the tangent bundle, i.e. $GL^+(n,\mathbb R)$, to the group of linear conformal authomorphisms of $\mathbb R^2$, i.e. $Aff(\mathbb R^2)$. Thus a conformal structure is a section of the $GL^+(n,\mathbb R)/Aff(\mathbb R^2)$-bundle that is associated with the tangent bundle $TM$. The space $GL^+(n,\mathbb R)/Aff(\mathbb R^2)$ can be identified with a unit disk $D$ in $\mathbb R^2$. In our case $M=\mathbb R^2$ is contractible, so the bundle is trivial. Thus conformal structures (induced by the given orientation) are simply smooth maps from $\mathbb R^2$ to $D$, i.e. $C^\infty(\mathbb R^2, D)$. The Teichmuller space is the quotient of this space by the $Diff_0(\mathbb R^2)$-action given by precomposing with diffeomorphisms.