# Quasi-Fuchsian 3-Manifolds and Metrics on Teichm\"{u}ller Space

## TL;DR

This paper studies almost Fuchsian 3-manifolds, showing how their minimal surfaces relate to paths in Teichmüller space, providing bounds on Teichmüller distances and introducing a new potential for the Weil-Petersson metric.

## Contribution

It establishes a connection between minimal surfaces in quasi-Fuchsian manifolds and Teichmüller space, including bounds on distances and a new potential for the Weil-Petersson metric.

## Key findings

- The path in Teichmüller space joins the conformal structures of the boundary components.
- An upper bound for the Teichmüller distance between boundary components is derived.
- A new potential function for the Weil-Petersson metric is introduced.

## 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 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.