On Lipschitz Normally Embedded singularities

TL;DR

This paper surveys Lipschitz Normally Embedded singularities, exploring their properties, criteria, examples, and open questions, with a focus on complex surfaces and their metric equivalences.

Contribution

It provides a comprehensive overview of Lipschitz Normally Embedded singularities, highlighting recent developments and open problems in the field.

Findings

Characterization criteria for Lipschitz Normally Embedded singularities

Examples of complex surfaces exhibiting Lipschitz Normal Embedding

Open questions guiding future research in the area

Abstract

Any subanalytic germ $(X,0) \subset (\mathbb R^n,0)$ is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ $(X,0)$. The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade. In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces. We conclude the manuscript with a list of open questions which we believe to be worth of interest.