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1 April 2019 Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume
Martin Bridgeman, Jeffrey Brock, Kenneth Bromberg
Duke Math. J. 168(5): 867-896 (1 April 2019). DOI: 10.1215/00127094-2018-0061

Abstract

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.

Citation

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Martin Bridgeman. Jeffrey Brock. Kenneth Bromberg. "Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume." Duke Math. J. 168 (5) 867 - 896, 1 April 2019. https://doi.org/10.1215/00127094-2018-0061

Information

Received: 19 October 2017; Revised: 15 November 2018; Published: 1 April 2019
First available in Project Euclid: 2 March 2019

zbMATH: 07055195
MathSciNet: MR3934591
Digital Object Identifier: 10.1215/00127094-2018-0061

Subjects:
Primary: 30F40
Secondary: 37F30

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 5 • 1 April 2019
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