The Fenchel-Nielsen Deformation

Abstract

The uniformization theorem provides that a Riemann surface S of negative Euler characteristic has a metric of constant curvature -1. A hyperbolic structure can be understood in terms of its deformations. Unfortunately, the variation of the hyperbolic metric, arising from a deformation, is not determined by local data. If z is a conformal coordinate for S and A, II H II 00 < 1, a Beltrami differential then dw = dz + p dJ defines a new conformal coordinate and structure SI. The tensor p may vanish on the open set 0 C S and yet the hyperbolic metrics of S and SI do not coincide on (. This phenomenon presents a s~erious difficulty in the deformation theory of hyperbolic structures. The Fenchel-Nielsen (twist) deformation does not share this defect. The deformation is defined by cutting the surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching the sides in their new position. A geodesic intersecting the cut is deformed to a broken geodesic. The hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface. The deformation is concentrated at the cut. The basic idea for the deformation appears in the work of Fricke-Klein, Dehn, and Fenchel-Nielsen [71, [8]. In the Fenchel-Nielsen manuscript the deformation was considered extensively; coordinates for Teichmiller space are defined in terms of the deformation. The proper definition is given by a Fuchsian group construction. Calculations are made in terms of unimodular matrices. This approach is often cumbersome. Our basic objective is to give a description of the deformation in terms of quasiconformal mappings. The Bers embedding of Teichmiiller space is central to the analytic theory of deformations. Using our characterization we calculate the first variation of the Bers embedding for the Fenchel-Nielsen deformation. The result is a Poincare series 6*, which already appears in the work of Petersson [16].