We consider a natural nonnegative two-form on quasifuchsian space that extends the Weil–Petersson metric on Teichmüller space. We describe completely the positive definite locus of , 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 is equal to the pullback of the pressure metric from dynamics. We use the properties of 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.
"Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space." Geom. Topol. 14 (2) 799 - 831, 2010. https://doi.org/10.2140/gt.2010.14.799