  We notice that a generic nonsingular gradient field $v = \nabla f$ on a
compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$
of $X$. We study the transformations of $K(f, v)$ that are induced by
deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$
with counting the \emph{double-tangent} trajectories of the $v$-flow, i.e. the
trajectories that are tangent to the boundary $\d X$ at a pair of distinct
points. Let $gc(X)$ be the minimum number of such trajectories, minimum being
taken over all nonsingular $v$'s. We call $gc(X)$ the \emph{gradient
complexity} of $X$. Next, we prove that there are only finitely many $X$ of
bounded gradient complexity, provided that $X$ is irreducible and boundary
irreducible with no essential annuli. In particular, there exists only finitely
many hyperbolic manifolds $X$ with bounded $gc(X)$. For such $X$, their
normalized hyperbolic volume gives an upper bound of $gc(X)$. If an irreducible
and boundary irreducible $X$ with no essential annuli admits a nonsingular
gradient flow with no double-tangent trajectories, then $X$ is a standard ball.
All these and many other results of the paper rely on a careful study of the
stratified geometry of $\d X$ relative to the $v$-flow. It is characterized by
failure of $\d X$ to be \emph{convex} with respect to a generic flow $v$. It
turns out, that convexity or its lack have profound influence on the topology
of $X$. This interplay between intrinsic concavity of $\d X$ with respect to
any gradient-like flow and complexity of $X$ is in the focus of the paper.
