In this paper, we use the Weil-Petersson gradient flow for renormalized
volume to study the space CC(N;S,X) of convex cocompact hyperbolic structures
on the relatively acylindrical 3-manifold (N;S). Among the cases of interest
are the deformation space of an acylindrical manifold and the Bers slice of
quasi-Fuchsian space associated to a fixed surface. To treat the possibility of
degeneration along flow-lines to peripherally cusped structures, we introduce a
surgery procedure to yield a surgered gradient flow that limits to the unique
structure MgeodββCC(N;S,X) with totally geodesic convex core
boundary facing S. Analyzing the geometry of structures along a flow line, we
show that if VRβ(M) is the renormalized volume of M, then
VRβ(M)βVRβ(Mgeodβ) is bounded below by a linear function of the
Weil-Petersson distance dWPβ(βcβM,βcβMgeodβ),
with constants depending only on the topology of S. The surgered flow gives a
unified approach to a number of problems in the study of hyperbolic
3-manifolds, providing new proofs and generalizations of well-known theorems
such as Storm's result that Mgeodβ has minimal volume for N
acylindrical and the second author's result comparing convex core volume and
Weil-Petersson distance for quasifuchsian manifolds