Ambient Lipschitz equivalence of real surface singularities

TL;DR

This paper constructs examples of real surface singularities that are bi-Lipschitz equivalent and topologically the same but differ in their ambient Lipschitz structure, revealing nuanced geometric distinctions.

Contribution

It introduces new examples demonstrating that outer bi-Lipschitz equivalence does not imply ambient Lipschitz equivalence for real surface singularities.

Findings

Existence of pairs of singular surfaces with different ambient Lipschitz types

Construction of infinitely many non-equivalent surfaces for each singularity

Distinction between bi-Lipschitz and ambient Lipschitz equivalences

Abstract

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset {\mathbb R}^4$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to $S$, but pairwise ambient Lipschitz non-equivalent.