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Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Martin Bridgeman

Geometry & Topology 14 (2010) 799–831

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.

quasifuchsian space, Weil–Petersson metric, Hausdorff dimension
Mathematical Subject Classification 2000
Primary: 30F60, 30F40, 37D35
Received: 9 February 2009
Revised: 3 January 2010
Accepted: 29 December 2009
Published: 2 March 2010
Proposed: Shigeyuki Morita
Seconded: Benson Farb, Martin Bridson
Martin Bridgeman
Department of Mathematics
Boston College
Chestnut Hill, MA 02167