Complex earthquakes and Teichmüller theory

It is known that any two points in Teichmüller space are joined by an earthquake path. In this paper we show any earthquake path $\mathbb R \rightarrow T(S)$ extends to a proper holomorphic mapping of a simply-connected domain $D$ into Teichmüller space, where $\mathbb R \subset D \subset \mathbb C$. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1-dimensional Teichmüller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.