Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

Abstract

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of (−1, 1). By the work of Uhlenbeck, such a 3-manifold M admits a foliation of parallel surfaces, whose locus in Teichmuller space is represented as a path γ, we show that γ joins the conformal structures of the two components of the conformal boundary of M . Moreover, we obtain an upper bound for the Teichmuller distance between any two points on γ, in particular, the Teichmuller distance between the two components of the conformal boundary of M , in terms of the principal curvatures of the minimal surface in M . We also establish a new potential for the Weil-Petersson metric on Teichmuller space.