I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to biholomorphism. How about hyperbolic structures (i.e. complete Riemannian metrics with constant sectional curvature $-1$)?
My general understanding of Teichmüller theory (at least for closed surfaces) is that the Teichmüller space $\mathcal{T}(S_{g})$ is the universal (orbifold) covering space of the Moduli space $\mathcal{M}(S_{g})$ and the covering projection is given by the action of the Mapping class group $Mod(S_{g})$. I don't know if this picture is preserved for non closed surfaces.
Maybe I should express my main doubts in two questions:
do we have different hyperbolic structures on $D$? can you provide an example of two non equivalent hyperbolic structures on $D$ (or on the upper-half plane if it's simpler)? I would be happy at least with a "guideline" to distinguish two structures: for example on a closed surface I can distinguish hyperbolic structures via the length of closed geodesics; but there's not such a thing in $D$.
If the answer to the previous question is positive, how can I reconcile the general picture sketched above for closed surfaces $Mod(S_{g}) \curvearrowright \mathcal{T}(S_{g}) \to \mathcal{M}(S_{g})$ in the case of the unit disk, where both $Mod(D)$ and $\mathcal{M}(D)$ are trivial?
Thanks in advance for any comments/suggestion/reference...
What you need is the following theorem:
Theorem. Let $(M,g)$ be a simply-connected complete Riemannian $n$-dimensional manifold of the sectional curvature $-1$. Then $(M,g)$ is isometric to the hyperbolic $n$-space.
Similar theorems hold for manifolds of sectional curvature equal to $0$ and to $+1$. You can find a proof in do Carmo's "Riemannian Geometry", which is my favorite introduction to the subject.
Of course, if you drop the completeness requirement, this theorem will be false.