Universality theorem for LNE H\"older triangles

TL;DR

This paper investigates the relationship between ambient and outer Lipschitz geometry of Lipschitz normally embedded H"older triangles in four-dimensional space, revealing infinitely many equivalence classes linked to microknots, unlike the three-dimensional case.

Contribution

It establishes a universality theorem demonstrating the complexity of Lipschitz geometry for H"older triangles in four dimensions, highlighting the role of microknots.

Findings

Infinitely many Lipschitz equivalence classes in $\

Microknots are central to classifying these equivalence classes.

Contrast with the three-dimensional case where classes are finite.

Abstract

We compare ambient and outer Lipschitz geometry of Lipschitz normally embedded H\"older triangles in $\mathbb{R}^4$. In contrast to the case of $\mathbb{R}^3$ there are infinitely many equivalence classes. The equivalence classes are related to the so-called microknots.