Geometry & Topology

Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Martin Bridgeman

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732207

Digital Object Identifier
doi:10.2140/gt.2010.14.799

Mathematical Reviews number (MathSciNet)
MR2602852

Zentralblatt MATH identifier
1200.30037

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

Keywords
quasifuchsian space Weil–Petersson metric Hausdorff dimension

Citation

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. https://projecteuclid.org/euclid.gt/1513732207


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