Asian Journal of Mathematics

Volume 23 (2019)

Number 6

Bi-Lipschitz geometry of contact orbits in the boundary of the nice dimensions

Pages: 953 – 968

Authors

Saurabh Trivedi (Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brazil; and the Indian Statistical Institute, North-East Centre, Assam, India)

Maria Aparecida Soares Ruas (Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brazil )

Abstract

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.

Keywords

bi-Lipshitz contact equivalence, Thom–Mather stratification, unimodular strata, boundary of the nice dimensions, Thom–Levine lemma

2010 Mathematics Subject Classification

14B05, 32S15, 58K40

The first author was supported by FAPESP grant 2014/00304-2 and CNPq grant 306306/2015-8. The second author was supported by FAPESP grant 2015/12667-5.

Received 2 July 2018

Accepted 28 September 2018