Ambient Lipschitz geometry of normally embedded surface germs

Abstract

We study the ambient Lipschitz geometry of semialgebraic surfaces. It was discovered in \cite{BBG} that ambient Lipschitz Geometry is different from the outer Lipschtz geometry. We show that two surface germs in ${\mathbb{R}}^3$, Lipschitz normally embedded and with isolated singularity, are ambient bi-Lipschitz equivalent if, and only if, they are outer bi-Lipschitz equivalent and ambient topologically equivalent.

64 pages, 38 figures. In the second version, several typos were corrected and some proofs were optimized. We also added a section devoted to prove the main result for several connected components with isolated singularity, showing also a counterexample with non-isolated singularity