2006/10/31 by Gabriel Katz, Katz, Gabriel
Mathematics · #57M20 #57M27 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M20 #msc:57M27
paper · pdf · doi:10.48550/arxiv.math/0611005
60 pages, 35 figures, pdfsync.sty, graphix.sty
arxiv created 2008/02/19 · arxiv updated 2009/12/01
We notice that a generic nonsingular gradient field v = ∇ 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 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 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 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.