BI-LIPSCHITZ GEOMETRY OF CONTACT ORBITS IN THE BOUNDARY OF THE NICE DIMENSIONS

摘要

Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are 'nice dimensions' while topological stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall C-1-stability is also a generic condition precisely in the nice dimensions. We address the question of bi-Lipschitz stability in this article. We prove that the Thom-Mather stratification is bi-Lipschitz contact invariant in the boundary of the nice dimensions. This is done in two steps: first we explicitly write the contact unimodular strata in every pair of dimensions lying in the boundary of the nice dimensions and second we construct Lipschitz vector fields whose flows provide the bi-Lipschitz contact trivialization in each of the cases.