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An extension of the Weil–Petersson metric to quasi-Fuchsian space

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Abstract

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmüller space. We use this to describe a metric on Teichmüller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson metric of the associated earthquake vector of the lamination.

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02167, USA

    Martin J. Bridgeman

  2. Department of Mathematics, Wesleyan University, Middletown, CT, 06459, USA

    Edward C. Taylor

Authors
  1. Martin J. Bridgeman
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  2. Edward C. Taylor
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Correspondence to Martin J. Bridgeman.

Additional information

Martin Bridgeman research supported in part by NSF grant DMS 0305634. Edward C. Taylor research supported in part by NSF grant DMS 0305704.

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Bridgeman, M.J., Taylor, E.C. An extension of the Weil–Petersson metric to quasi-Fuchsian space. Math. Ann. 341, 927–943 (2008). https://doi.org/10.1007/s00208-008-0218-3

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  • DOI: https://doi.org/10.1007/s00208-008-0218-3

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