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.
Best Answer
Although applications to Thurston's work on 3-manifolds and Bers' embedding has already been mentioned, I thought it was worth mentioning Bers' simultaneous uniformization theorem (underlying Bers' embedding), which implies that for any two points in Teichmuller space, there is a unique (marked) quasifuchsian surface, such that the two conformal structures on the domain of discontinuity realize both Riemann surfaces simultaneously. This result was generalized by Ahlfors, Bers, and Sullivan, who showed more generally that the space of geometrically finite Kleinian groups are parameterized by the Teichmuller spaces of the conformal structures on the domain of discontinuity. This is the starting point for the classification of Kleinian groups (and Thurston's work on the geometrization of Haken 3-manifolds).
Related to this is the parameterization of a complex projective structure on a surface by a conformal structure and a holomorphic quadratic differential, so that the projective structures are in bijection with the cotangent space to Teichmuller space.
A related result of Bers gave Thurston's classification of mapping classes of surfaces into pseudo-Anosov, reducible, or finite order, by minimizing the translation length of a mapping class in Teichmuller space.
Teichmuller space has also been used as an ingredient in the construction of various (projective) representations of mapping class groups, notably by Andersen based on the Hitchin connection (realizing ideas of Witten).
In another direction, Labourie and Loftin discovered that convex projective structures on a surface are parameterized by a conformal structure together with a cubic differential, so may be regarded as a bundle over Teichmuller space.