Classification of Lipschitz simple function germs

TL;DR

This paper provides a complete classification of Lipschitz simple function germs in the complex case, revealing that Lipschitz modal germs deform to a specific singularity family called $J_{10}$.

Contribution

It introduces the concept of Lipschitz simple function germs and offers a full classification in the complex setting, connecting Lipschitz modality to Arnold's singularity list.

Findings

Lipschitz modal germs deform to the $J_{10}$ family.

Lipschitz simple germs are fully classified in the complex case.

A new notion of Lipschitz simplicity is introduced.

Abstract

It is known that the bi-Lipschitz right classification of function germs admit moduli. In this article we introduce a notion called the Lipschitz simple function germs and present a full classification in the complex case. A surprising consequence of our result is that a function germ is Lipschitz modal if and only if it deforms to the smooth unimodal family of singularities called $J_{10}$ in Arnold's list.