The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

Brock, Jeffrey¹

¹ Mathematics Department, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Journal of the American Mathematical Society, Tome 16 (2003) no. 3, pp. 495-535

Résumé

We present a coarse interpretation of the Weil-Petersson distance $d_{\mathrm {WP}}(X,Y)$ between two finite area hyperbolic Riemann surfaces $X$ and $Y$ using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold $Q(X,Y)$ with $X$ and $Y$ in its conformal boundary is comparable to the Weil-Petersson distance $d_{\mathrm {WP}}(X,Y)$. In applications, we relate the Weil-Petersson distance to the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian for $Q(X,Y)$, and give a new finiteness criterion for geometric limits.

Détail

DOI : 10.1090/S0894-0347-03-00424-7

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