A metric uniformization model for the quasi-Fuchsian space

We introduce and study a novel metric uniformization model for the quasi-Fuchsian space Q F (S) QF (S), defined through a class of C C -valued bilinear forms on S S, called Bers metrics, which coincide with hyperbolic Riemannian metrics along the Fuchsian locus. By employing this approach, we present a new model of the holomorphic tangent bundle of Q F (S) QF (S) that extends the metric model for the Teichmüller space defined by Berger and Ebin, and give an integral representation of the Goldman symplectic form and of the holomorphic extension of the Weil-Petersson metric to Q F (S) QF (S), with a new proof of its existence and non-degeneracy. We also determine new bounds for the Schwarzian of Bers projective structures extending Kraus’ estimate. Lastly, we use this formalism to give alternative proofs to several classic results in quasi-Fuchsian theory.