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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 Teichmuller theory (at least for closed surfaces) is that the Teichmuller space $\mathcal{T}(S_{g,n})$ is the universal (orbifold) covering space of the Moduli space $\mathcal{M}(S_{g,n})$ and the covering projection is given by the action of the Mapping class group $Mod(S_{g,n})$. I don't know if this picture is preserved for non closed surfaces.

Maybe I should express my main doubts in two questions:

  1. do we have different hyerbolic 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 $D$.

  2. If the answer to the previous question is positive, how can I reconcile the general picture sketched above for closed surfaces $Mod(S_{g,n}) \curvearrowright \mathcal{T}(S_{g,n}) \to \mathcal{M}(S_{g,n})$ 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...

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1 Answer 1

up vote 3 down vote accepted

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.

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So of course there is just one structure. What is the right kind of hyperbolic structure to look at to see some flexibility? I think I am looking for relationships with the so called "infinite Teichmuller space". –  Lor Jun 2 at 18:00
It is the same hyperbolic structure, just given an extra data (a quasisymmetric map at infinity). Lehto's book "Univalent functions and Teichmüller spaces" might be a good place to start. –  studiosus Jun 2 at 18:02
I see. So this extra data (quasisymmetric map) does not encode anything about the complex/hyperbolic structure of $D$, but is rather an artificial tool used to build a universal object among Teichmuller spaces, right? –  Lor Jun 2 at 18:12
Yes, except I would call it not "artifical", but "auxiliary". It is like in the ordinary Teichmuller theory, the main object is not just a compact Riemann surface, but one together with an auxiliary object, a homotopy class of maps from a reference Riemann surface. –  studiosus Jun 2 at 18:15

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