  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.
