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

TL;DR

This paper establishes a combinatorial framework linking Weil-Petersson distances between hyperbolic surfaces to the volumes of convex cores in associated 3-manifolds, revealing new geometric and spectral connections.

Contribution

It introduces a coarse combinatorial description of Weil-Petersson distance and demonstrates its relation to convex core volumes in quasi-Fuchsian 3-manifolds, advancing understanding of hyperbolic geometry.

Findings

Weil-Petersson distance is comparable to convex core volume.

Connections established between Weil-Petersson distance, limit set dimension, and Laplacian eigenvalues.

Provides a new finiteness criterion for geometric limits.

Abstract

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.