On the bi-Lipschitz contact equivalence of plane complex function-germs

TL;DR

This paper investigates the bi-Lipschitz contact equivalence of complex plane function-germs, revealing that their infinitesimal size classification aligns with the right topological classification, thus connecting geometric and topological perspectives.

Contribution

It establishes the equivalence between bi-Lipschitz contact classification and right topological classification for complex plane function-germs.

Findings

Bi-Lipschitz contact equivalence matches right topological classification.

Infinitesimal size of function-germs is characterized up to bi-Lipschitz changes.

The problem reduces to a topological classification problem.

Abstract

In this short note, we consider the problem of bi-Lipschitz contact equivalence of complex analytic function-germs of two variables. It is inquiring about the infinitesimal sizes of such function-germs, up to bi-Lipschitz changes of coordinates. We show that this problem is equivalent to the problem of the right topological classification.