Lower bounds on volumes of hyperbolic Haken 3-manifolds

Abstract

We prove a volume inequality for 3-manifolds having <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 0"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> metrics “bent” along a surface and satisfying certain curvature conditions. The result makes use of Perelman’s work on the 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 of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.