# The space of conformal structures on the Euclidean plane

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.

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As far as I know, there are only two possible conformal structures on $R^2$, given by the flat and hyperbolic metrics. –  Deane Yang Sep 8 '11 at 17:04
@Deane: sure, but this does not answer my question. To give you an idea of what I want, recall that any metric on $S^2$ is conformal to the standard curvature $1$ metric, yet Earle-Eells showed in their 1969 JDG paper that the space of conformal structures on $S^2$ is homeomorphic to the group of self-diffeomorphisms of $S^2$ that are isotopic to identity and fix $3$ points. I do not know a similar result for $\mathbb R^2$, where these is also an issue of what topology to put on the space (e.g. compact-open or uniform). –  Igor Belegradek Sep 8 '11 at 17:23
If you know it for $\mathbb S^2$ then you know it for flat $\mathbb R^2$ which is $\mathbb S^2$ without one point. Did I miss something? –  Anton Petrunin Sep 8 '11 at 17:57
Anton: I know little of these matters (which is why I ask), but is what you are saying a general phonomena? E.g. does knowing the space of conformal structures for the torus is enough to to understand the same space for the punctured torus. This seems unlikely. –  Igor Belegradek Sep 8 '11 at 18:07
Igor, thanks. I guess I don't know what the "space of conformal structures" is. I'll have to look at the Earle-Eells paper. –  Deane Yang Sep 8 '11 at 18:54