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 Teichmüller space. We use this to describe a metric on Teichmüller 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.

Thermodynamics, dimension and the Weil–Petersson metric

“…which is sharpened by Theorem 1.1, and include an inequality version of Corollary 1.2 [BT2,§11]. We remark that the pseudometric on quasifuchsian space introduced in [BT2] can also be regarded as a pullback of the pressure metric.…”

“…We remark that the pseudometric on quasifuchsian space introduced in [BT2] can also be regarded as a pullback of the pressure metric. I would also like to thank M. Zinsmeister and the referees for many useful comments.…”

Simple length rigidity for Kleinian surface groups and applications

“…Burger [9] introduced a renormalized pressure intersection between convex cocompact representations into rank one Lie groups. Bridgeman and Taylor [7] extensively studied this renormalized pressure intersection in the setting of quasifuchsian representation. We say that ρ : π 1 (S) → PSL(2, C) is quasifuchsian if it is topologically conjugate, in terms of its action on C, to a Fuchsian representation into PSL(2, R).…”

“…Bridgeman and Taylor [7] showed that the Hessian of J gives rise to a non-negative bilinear form on the tangent spaceTQF (S) of quasifuchsian space, called the pressure form. Motivated by work of McMullen [24] in the setting of Teichmüller space, Bridgeman [5] used the thermodynamic formalism to show that the only degenerate vectors for the pressure form correspond to pure bending at points on the Fuchsian locus.…”

An introduction to pressure metrics for higher Teichmüller spaces

“…Bridgeman and Taylor [18] used Patterson-Sullivan theory to show that the Hessian of the renormalized intersection number is a non-negative form on quasifuchsian case. McMullen [62] then introduced the use of the techniques of Thermodynamic Formalism to interpret both of these metrics as pullbacks of the pressure metric on the space of suspension flows on the shift space associated to the Bowen-Series coding.…”