## Geometry & Topology

### Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Martin Bridgeman

#### 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

#### 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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