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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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