Title: 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 $\|\phi_\Sigma\|_\infty$ 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 $N$.