Complex earthquakes and Teichmüller theory

McMullen, Curtis

doi:10.1090/S0894-0347-98-00259-8

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Complex earthquakes and Teichmüller theory

Author: Curtis T. McMullen
Journal: J. Amer. Math. Soc. 11 (1998), 283-320
MSC (1991): Primary 30F10, 30F40, 32G15
MathSciNet review: 1478844
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Abstract: 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.

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