Duke Mathematical Journal

Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

Martin Bridgeman, Jeffrey Brock, and Kenneth Bromberg

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To a complex projective structure Σ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms ϕΣ and ϕΣ2 of the quadratic differential ϕΣ of Σ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space CC(N) of convex cocompact hyperbolic structures on a compact manifold N with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of DN, the double of N.

Article information

Duke Math. J., Volume 168, Number 5 (2019), 867-896.

Received: 19 October 2017
Revised: 15 November 2018
First available in Project Euclid: 2 March 2019

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Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems

renormalized volume Schwarzian derivative hyperbolic geometry Kleinian groups Weil–Petersson metric


Bridgeman, Martin; Brock, Jeffrey; Bromberg, Kenneth. Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume. Duke Math. J. 168 (2019), no. 5, 867--896. doi:10.1215/00127094-2018-0061. https://projecteuclid.org/euclid.dmj/1551495708

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