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

Abstract

We present a coarse interpretation of the Weil-Petersson distance d W P ( X , Y ) d_{\mathrm {WP}}(X,Y) between two finite area hyperbolic Riemann surfaces X X and Y 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 ) Q(X,Y) with X X and Y Y in its conformal boundary is comparable to the Weil-Petersson distance d W P ( X , Y ) 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 ) Q(X,Y) , and give a new finiteness criterion for geometric limits.