Lipschitz Geometry of Real Semialgebraic Surfaces

TL;DR

This paper reviews recent advances in Lipschitz geometry of semialgebraic surface germs, highlighting the differences between inner, outer, and ambient classifications, and their connections to knot theory.

Contribution

It clarifies the complexity differences between inner, outer, and ambient Lipschitz classifications and discusses the inclusion of knot theory in ambient Lipschitz geometry.

Findings

Inner Lipschitz classification of surface germs is solved.

Outer Lipschitz classification remains open and is more complex.

Ambient Lipschitz geometry encompasses all of knot theory.

Abstract

We present here basic results in Lipschitz Geometry of semialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer metric remains an open problem. We review recent results related to the outer and ambient bi-Lipschitz classification of surface germs. In particular, we explain why the outer Lipschitz classification is much harder than the inner classification, and why the ambient Lipschitz Geometry of surface germs is very different from their outer Lipschitz Geometry. In particular, we show that the ambient Lipschitz Geometry of surface germs includes all of the Knot Theory.