Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space

TL;DR

This paper introduces a new metric G on quasifuchsian space extending the Weil-Petersson metric, analyzes its properties, and uses it to show that the Hausdorff dimension function has no local maxima in this space.

Contribution

It defines and characterizes a natural non-negative two-form G extending the Weil-Petersson metric and relates it to the pressure metric, providing new insights into the geometry of quasifuchsian space.

Findings

G is positive definite off the fuchsian diagonal

G equals the pullback of the pressure metric

Hausdorff dimension has no local maxima in quasifuchsian space

Abstract

We consider a natural non-negative two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmuller space. We describe completely the positive definite locus of G, showing that it is a positive definite metric off the fuchsian diagonal of quasifuchsian space and is only zero on the "pure-bending'' tangent vectors to the fuchsian diagonal . We show that G is equal to the pullback of the pressure metric from dynamics. We use the properties of G to prove that at any critical point of the Hausdorff dimension function on quasifuchsian space the Hessian of the Hausdorff dimension function must be positive definite on at least a half-dimensional subspace of the tangent space. In particular this implies that Hausdorff dimension has no local maxima on quasifuchsian space.