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

## TL;DR

This paper establishes bounds on the geometry of complex projective structures using Schwarzian derivatives, generalizes known hyperbolic structure results, and studies the Weil-Petersson gradient flow of renormalized volume, proving a key conjecture.

## Contribution

It introduces a unified approach to bounding geometric features of projective structures and analyzes the Weil-Petersson gradient flow, proving the minimality of renormalized volume.

## Key findings

- Derived bounds on geometry in terms of quadratic differential norms
- Generalized bounds for convex cocompact hyperbolic structures
- Proved the conjecture on the infimum of 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 $\|\phi_\Sigma\|_\infty$ and $\|\phi_\Sigma\|_2$ of the quadratic differential $\phi_\Sigma$ 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 $N$.