Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings

TL;DR

This paper investigates metric measure spaces with rich families of curves, showing they cannot embed into Euclidean spaces unless they exhibit an infinitesimal splitting structure, with implications for conformal dimension and non-embedding proofs.

Contribution

It establishes a new necessary condition called infinitesimal splitting for bi-Lipschitz embeddability of spaces with thick curve families into Euclidean spaces.

Findings

Spaces with thick curve families cannot embed into Euclidean spaces without infinitesimal splitting.

Introduces the concept of infinitesimal splitting as a key geometric property.

Provides new proofs for known non-embedding results and applications to conformal dimension.

Abstract

We study metric measure spaces that admit "thick" families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some "infinitesimal splitting": their tangent spaces are bi-Lipschitz equivalent to product spaces of the form $Z\times \mathbb{R}^k$ for some $k\geq 1$. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.