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arXiv · 1002.1900

Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space

Martin Bridgeman
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Abstract

We consider a natural non-negative two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmuller 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. 2010-02-09. Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space. https://doi.org/10.2140/gt.2010.14.799

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