  We prove a volume inequality for 3-manifolds having C^0 metrics "bent" along
a hypersurface, and satisfying certain curvature pinching conditions. The
result makes use of Perelman's work on Ricci flow and geometrization of closed
3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about
volumes of convex cores of Kleinian groups, improved volume estimates for
certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume
orientable hyperbolic 3-manifold. An appendix by Dunfield compares estimates of
volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with
experimental data.
