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 extends to a proper holomorphic mapping of a simply-connected domain into Teichmüller space, where . 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.