I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:
- No-wandering-domain Theorem (Sullivan)
- Theorems of Thurston (Classification of homeomorphism of surfaces, topological characterization of rational maps, hyperbolization theorems for special 3 manifolds)
- Computer graphics. (Using the various metrics on the Teichmüller spaces as a substitute for Gromov-Hausdorff metric.)
- String Theory (as elementary particles are modeled by loops, they generate a Riemann surface as they move through time.)
- Some applications to biology.(Brain morphometry)
Note: I am very sure that this is only a small fraction of what is out there, and I plan to continue to update this list.