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Complex Earthquakes and Teichmuller Theory

It is known that any two points in Teichmuller space are joined by an earthquake path. In this paper we show any earthquake path R → T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmuller space, where R ⊂ D ⊂ 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 Teichmuller 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.