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 $DN$, the double of $N$.

20 pages, 0 figures

References in corpus (1)

• On the renormalized volume of hyperbolic 3-manifolds

Cited by in corpus (6)

• No Ensemble Averaging Below the Black Hole Threshold
• The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
• The dual Bonahon-Schläfli formula
• A new uniform lower bound on Weil-Petersson distance
• CMC-1 surfaces via osculating Möbius transformations between circle patterns
• The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds