Abstract: We notice that a generic
nonsingular gradient field on a compact 3-fold Xwith boundary canonically
generates a simple spine of X. We study the
transformations of that are induced by deformations of the data We link the Matveev complexity of Xwith counting the double-tangenttrajectories of the v-flow, i.e., the trajectories that are tangent to the boundary at a pair of distinct points. Let be the minimum number of such trajectories, minimum being
taken over all nonsingular v’s. We call the gradient complexityof X. Next, we prove that there are only finitely many Xof bounded gradient
complexity, provided that Xis irreducible and has no essential annuli. In particular, there exists
only finitely many hyperbolic manifolds Xwith bounded For such X, their normalized
hyperbolic volume gives a lower bound of If an orientable X withadmits a non-singular
gradient flow with one double-tangent trajectory at most, then Xis a connected sum of
several 3-balls and products All these and many other results of the paper rely on a
careful study of the stratified geometry of relative to the v-flow. It is characterized
by failure of to be convexwith 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 with respect to any
gradient-like flow and the complexity is in the focus of the paper.
Keywords and phrases: gradient flows, spines, complexity of 3-folds.