Complex Earthquakes and Teichmuller Theory

Abstract

It is known that any two points in Teichmuller 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 simplyconnected domain D into Teichmuller space, where \(\mathbb{R} ⊂ \mathbb{D} ⊂ \mathbb{C}\). 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 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.