A metric uniformization model for the quasi-Fuchsian space

TL;DR

This paper introduces a new uniformization metric model for the quasi-Fuchsian space of a surface, extending classical structures and providing new integral representations and bounds, with applications to classic quasi-Fuchsian results.

Contribution

It presents a novel metric model for QF(S) using Bers metrics, extending the tangent bundle model of Teichmüller space and offering new proofs and bounds in quasi-Fuchsian theory.

Findings

Integral representation of Goldman symplectic form

Extension of Weil-Petersson metric to QF(S)

New bounds for the Schwarzian of Bers projective structures

Abstract

We introduce and study a novel uniformization metric model for the quasi-Fuchsian space QF(S) of a closed oriented surface S, defined through a class of C-valued bilinear forms on 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 QF(S) that extends the metric model for Teichm\"uller 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 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.