## Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume.(English)Zbl 1420.32007

Summary: 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 \(\operatorname{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$$

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F60 Teichmüller theory for Riemann surfaces 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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