Translator Disclaimer
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


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.


Download Citation

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.


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

Primary: 30F40
Secondary: 37F30

Rights: Copyright © 2019 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.168 • No. 5 • 1 April 2019
Back to Top