Geometry & Topology

Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Martin Bridgeman

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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.

Article information

Geom. Topol., Volume 14, Number 2 (2010), 799-831.

Received: 9 February 2009
Revised: 3 January 2010
Accepted: 29 December 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15] 30F40: Kleinian groups [See also 20H10] 37D35: Thermodynamic formalism, variational principles, equilibrium states

quasifuchsian space Weil–Petersson metric Hausdorff dimension


Bridgeman, Martin. Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space. Geom. Topol. 14 (2010), no. 2, 799--831. doi:10.2140/gt.2010.14.799.

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