Lipschitz geometry and combinatorics of abnormal surface germs
Abstract
We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable H\"older triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal H\"older triangles.
Cite
@article{arxiv.2101.02302,
title = {Lipschitz geometry and combinatorics of abnormal surface germs},
author = {Andrei Gabrielov and Emanoel Souza},
journal= {arXiv preprint arXiv:2101.02302},
year = {2021}
}
Comments
58 pages, 14 figures. In Version 3, Examples 2.3, 2.21 and 2.46 added. Fig. 1 illustrating Example 2.21 added. Definition 2.43 of a perfect zone is modified. Lemma 4.7 and Fig. 7 illustrating its proof moved from Section 5 to Section 4. Proofs of Lemma 4.8 and Proposition 4.10 simplified, two figures illustrating previous proofs removed