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

TL;DR

This paper investigates bi-Lipschitz stability of contact orbits in the boundary of nice dimensions, proving invariance of stratification and constructing trivializations via bi-Lipschitz vector fields.

Contribution

It establishes bi-Lipschitz contact invariance of Thom-Mather stratification in the boundary of nice dimensions and provides explicit constructions for trivializations.

Findings

Bi-Lipschitz contact invariance of stratification proven.

Explicit description of contact unimodular strata in boundary of nice dimensions.

Construction of bi-Lipschitz vector fields for trivialization.

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 topologically 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 bi-Lipschitz vector fields whose flow provide the bi-Lipschitz contact trivialization in each of the cases.