Lipschitz geometry and combinatorics of abnormal surface germs

TL;DR

This paper explores the Lipschitz geometry of real surface germs, revealing that abnormal arcs form finitely many zones and are closely linked to the combinatorial structure of H"older triangles.

Contribution

It introduces a detailed analysis of abnormal arcs in Lipschitz geometry, connecting geometric properties with combinatorial aspects of surface germs.

Findings

Abnormal arcs form finitely many zones in the arc space.

A strong relation exists between geometry and combinatorics of abnormal H"older triangles.

Definable surface germs exhibit specific Lipschitz geometric behaviors.

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.