The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
Abstract
Description
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics 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 boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness criterion for geometric limits.
48 pages, 14 figures. Revised final version. To appear, J. Amer. Math. Soc
48 pages, 14 figures. Revised final version. To appear, J. Amer. Math. Soc