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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2023
- DOI:
- arXiv:
- arXiv:2311.18570
- Bibcode:
- 2023arXiv231118570B
- Keywords:
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- Mathematics - Metric Geometry;
- Mathematics - Algebraic Geometry;
- 14P10;
- 32V40
- E-Print:
- 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