Given a complex projective structure $\Sigma$ on a surface, Thurston associated 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. We show that these give a unifying approach that generalizes a number of well-known results for convex cocompact hyperbolic structures including bounds on the Lipschitz constant for the retract 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. This leads to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of $DN$, the double of $N$. 查看全文>>