Using simplicial volume to count multi-tangent trajectories of traversing vector fields

TL;DR

This paper establishes a lower bound on the number of multi-tangent boundary trajectories of certain vector fields on manifolds, using simplicial volume and Gromov's methods, with applications to hyperbolic manifolds.

Contribution

It introduces a novel lower bound relating boundary tangent trajectories to the simplicial volume of the manifold, extending Gromov's techniques.

Findings

Lower bound on multi-tangent trajectories in terms of simplicial volume

Application of bounds to hyperbolic manifolds

Extension of Gromov's amenable reduction lemma

Abstract

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds.