Lipschitz geometry of complex curves

TL;DR

This paper explores the Lipschitz geometry of complex curves, providing a stronger, more general proof that the outer Lipschitz geometry of a complex plane curve germ uniquely determines its embedded topology, extending known results.

Contribution

It offers a simplified, analytic-restriction-free proof that the outer Lipschitz geometry fully characterizes the embedded topology of complex plane curve germs.

Findings

Outer Lipschitz geometry determines embedded topology.

Proof does not rely on analytic restrictions.

Strengthens known results in Lipschitz geometry of complex curves.

Abstract

We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, but in an analytic category.