Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

TL;DR

This paper demonstrates that the outer Lipschitz geometry of complex surface singularities encodes significant analytic information and establishes equivalences between Lipschitz triviality and Zariski equisingularity for certain families.

Contribution

It proves that Lipschitz geometry determines analytic structure of complex surface singularities and establishes the equivalence between Lipschitz triviality and Zariski equisingularity in specific cases.

Findings

Lipschitz geometry determines analytic structure of surface singularities.

Constant Lipschitz geometry implies Zariski equisingularity.

Lipschitz triviality and Zariski equisingularity are equivalent for certain hypersurface families.

Abstract

We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.