Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Bridgeman, Martin¹

¹ Department of Mathematics, Boston College, Chestnut Hill, MA 02167

Geometry & topology, Tome 14 (2010) no. 2, pp. 799-831.

Résumé

We consider a natural nonnegative 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.

Détail

DOI : 10.2140/gt.2010.14.799

Keywords: quasifuchsian space, Weil–Petersson metric, Hausdorff dimension

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