Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space,”Geom

We consider a natural non-negative two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmüller 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. 1 Statement of results Let S be a closed hyperbolic surface and T (S) be the associated Teichmüller space. Then the Weil-Petersson metric w is a Riemannian metric on T (S). For simplicity, we normalize the Weil-Petersson metric to define the normal-ized Weil-Petersson metric g