Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Bridgeman, Martin

doi:10.2140/gt.2010.14.799

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Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Martin Bridgeman

Geometry & Topology 14 (2010) 799–831

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.

Keywords

quasifuchsian space, Weil–Petersson metric, Hausdorff dimension

Mathematical Subject Classification 2000

Primary: 30F60, 30F40, 37D35

References

Publication

Received: 9 February 2009

Revised: 3 January 2010

Accepted: 29 December 2009

Published: 2 March 2010

Proposed: Shigeyuki Morita

Seconded: Benson Farb, Martin Bridson

Authors

Martin Bridgeman
Department of Mathematics Boston College Chestnut Hill, MA 02167
http://www2.bc.edu/~bridgem/

Volume 14, issue 2 (2010)