An introduction to Lipschitz geometry of complex singularities

TL;DR

This paper introduces a new perspective on Lipschitz geometry of complex singularities, providing complete classifications for plane curve singularities and surface germs, and developing key tools like the bubble trick.

Contribution

It offers the first comprehensive classification of Lipschitz geometry for complex plane curves and normal surface singularities, introducing innovative methods such as the bubble trick.

Findings

Complete Lipschitz classification of complex plane curve singularities

Development of the bubble trick and bubble trick with jumps

Invariant geometric decompositions for surface germs

Abstract

The aim of this paper to introduce the reader to a recent point of view on the Lipschitz classifications of complex singularities. It presents the complete classification of Lipschitz geometry of complex plane curves singularities and in particular, it introduces the so-called bubble trick and bubble trick with jumps which are key tools to study Lipschitz geometry of germs. It describes also the thick-thin decomposition of a normal complex surface singularity and built two geometric decompositions of a normal surface germ into standard pieces which are invariant by respectively inner and outer bilipschitz homeomorphisms. This leads in particular to the complete classification of Lipschitz geometry for the inner metric.