Q:

Analogy of Liouville conformal mapping theorem with Mostow rigidity?

I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar:  they say that a set of geometric structures is, if nonempty, big in dimension 2, but small in dimension greater than 2.
(For Mostow's theorem, the set of structures in question is the set of hyperbolic metrics on a manifold; for Liouville's, it's the set of germs of flat metrics in a conformal equivalence class.)
I know that hyperbolic and conformal geometry are closely connected, at least in dimension 2.  I'm curious as to whether this analogy is hinting at one such connection.  Is there a "good reason" for this analogy?  

A:

There is a connection in some way: if I remember right, you usually prove Mostow rigidity by looking at the hyperbolic space, which is the universal cover of your hyperbolic manifold, then you consider its boundary, which is the flat conformal sphere. Any isometry of the hyperbolic space induces a conformal transformation of its boundary, and vice-versa. But I don't think that you can derive Mostow rigidity from Liouville's conformal theorem.

