Curvature of new Kähler metrics on the total space of Griffiths negative vector bundle and quasi-Fuchsian space

Abstract

We study Kähler metrics on the total space of Griffiths negative holomorphic vector bundles over Kähler manifolds. As an application, we construct mapping class group invariant Kähler metrics on ${ℬ}({𝒮})$, the holomorphic tangent bundle of Teichmüller space of a closed surface $S$. Consequently,we obtain a new mapping class group invariant Kähler metric on the quasi-Fuchsian space $\mathrm{\text{QF}}(S)$, which extends the Weil–Petersson metric on the Teichmüller space ${𝒯}(S)\subset \mathrm{\text{QF}}(S)$. We also calculate its curvature and prove non-positivity for the curvature along the tautological directions.