To a complex projective structure on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms and of the quadratic differential of 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 -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 of convex cocompact hyperbolic structures on a compact manifold with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of , the double of .
"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