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Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller 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 Teichmüller distance between any two points on $γ$, in particular, the Teichmüller 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 Teichmüller space.