New Kähler metric on quasifuchsian space and its curvature properties

Abstract

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\ge 2$. We construct a new mapping class group invariant Kähler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichmüller space $\mathcal{T}(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions.

14 pages