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

@article{Bridgeman2019SchwarzianDP,
  title={Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume},
  author={Martin Bridgeman and Jeffrey F. Brock and Ken Bromberg},
  journal={Duke Mathematical Journal},
  year={2019}
}
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… 
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