Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

Details Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

2009-09-13

arxiv.org/abs/0909.2426v2

Asian J. Math., 14 (2010), no. 2, 243-256

Mathematics

Differential Geometry

16 pages, minor revision, some typos are fixed

Scientific paper

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 Teichm\"{u}ller space is represented as a path $\gamma$, we show that $\gamma$ joins the conformal structures of the two components of the conformal boundary of $M$. Moreover, we obtain an upper bound for the Teichm\"{u}ller distance between any two points on $\gamma$, in particular, the Teichm\"{u}ller 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 Teichm\"{u}ller space.

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