  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 $M_{\rm geod} \in CC(N;S,X)$ with totally geodesic convex core
boundary facing $S$. Analyzing the geometry of structures along a flow line, we
show that if $V_R(M)$ is the renormalized volume of $M$, then
$V_R(M)-V_R(M_{\rm geod})$ is bounded below by a linear function of the
Weil-Petersson distance $d_{\rm WP}(\partial_c M, \partial_c M_{\rm geod})$,
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 $M_{\rm geod}$ has minimal volume for $N$
acylindrical and the second author's result comparing convex core volume and
Weil-Petersson distance for quasifuchsian manifolds.
