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 Hölder 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 Hölder triangles.