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

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

2001-09-06

arxiv.org/abs/math/0109048v2

Mathematics

Geometric Topology

48 pages, 14 figures. Revised final version. To appear, J. Amer. Math. Soc

Scientific paper

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.

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