Rigidity of Teichmuller Space

TL;DR

This paper proves the holomorphic rigidity of Teichmüller space using harmonic maps, showing the mapping class group action uniquely determines the space, and also provides new proofs of superrigidity results.

Contribution

It establishes the holomorphic rigidity of Teichmüller space via harmonic maps and offers alternative proofs for superrigidity of the mapping class group.

Findings

Singular set of harmonic maps has Hausdorff dimension at most n-2

Harmonic maps exhibit decay near singularities

Provides harmonic map proofs of superrigidity results

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.