Rigidity of Teichmuller Space

Abstract

We prove the holomorphic rigidity conjecture of Teichm\"{u}ller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichm\"{u}ller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth $n$-dimensional Riemannian domain to the Weil-Petersson completion $\overline{\mathcal T}$ of Teichm\"{u}ller space has Hausdorff dimension at most $n-2$, and moreover, $u$ has certain decay near the singular set. Combining this with the earlier work of Schumacher, Siu and Jost-Yau, we provide a proof of the holomorphic rigidity of Teichm\"{u}ller space. In addition, our results provide as a byproduct a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by Farb-Masur and Yeung.