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2010 Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space
Martin Bridgeman
Geom. Topol. 14(2): 799-831 (2010). DOI: 10.2140/gt.2010.14.799

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.

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Martin Bridgeman. "Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space." Geom. Topol. 14 (2) 799 - 831, 2010. https://doi.org/10.2140/gt.2010.14.799

Information

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

zbMATH: 1200.30037
MathSciNet: MR2602852
Digital Object Identifier: 10.2140/gt.2010.14.799

Subjects:
Primary: 30F40, 30F60, 37D35

Rights: Copyright © 2010 Mathematical Sciences Publishers

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Vol.14 • No. 2 • 2010
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