Schwarzian derivatives, projective structures, and the WeilPetersson 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 3manifolds, including bounds on the Lipschitz constant for the nearestpoint retraction and the length of the bending lamination. We then use these bounds to begin a study of the WeilPetersson 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 onehalf the simplicial volume of $DN$, the double of $N$.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.06021
 Bibcode:
 2017arXiv170406021B
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Geometric Topology;
 32G15;
 32Q45;
 51P05
 EPrint:
 20 pages, 0 figures