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 $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller 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 Teichmüller distance between any two points on $\gamma$, 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.

arXiv comment16 pages, minor revision, some typos are fixed

References in corpus (2)

• On the renormalized volume of hyperbolic 3-manifolds
• Foliations for Quasi-Fuchsian 3-Manifolds

Cited by in corpus (3)

• Embedded $H$-Planes in Hyperbolic 3-Space
• Curvatures on the Teichmüller curve
• Surfaces of Prescribed Mean Curvature in Quasi-Fuchsian Manifolds