An extension of the Weil–Petersson metric to quasi-Fuchsian space

Abstract

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmuller space. We use this to describe a metric on Teichmuller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson metric of the associated earthquake vector of the lamination.