2015/02/11 by Daskalopoulos, Georgios, Mese, Chikako
#32G15 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1502.03367
We prove the holomorphic rigidity conjecture of Teichmüller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmüller 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 \mathcal T of Teichmüller 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 Teichmüller 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.