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

2015/03/09 by Hannah Alpert, Alpert, Hannah, Gabriel Katz +1
Mathematics · #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1503.02583

Abstract

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

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