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

Abstract

To a complex projective structure $\Sigma$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\Vert {\varphi }_{\Sigma }{\Vert }_{\infty }$ and $\Vert {\varphi }_{\Sigma }{\Vert }_2$ of the quadratic differential ${\varphi }_{\Sigma }$ of $\Sigma$ 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 $\mathit{DN}$, the double of $N$.