论文标题
对于具有厚曲线家族和欧几里德嵌入的空间的无限分裂
Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
论文作者
论文摘要
我们以艾伯利表示或曲线为正模量的形式研究了通用曲线或曲线碎片的“厚”家族的度量度量空间。我们表明，除非他们承认某些“无限分裂”，否则不能将这些空间嵌入任何欧几里得空间中：它们的切线空间是bi-lipschitz等同于$ z \ times \ times \ times \ times \ mathbb {r}^k $的产品空间，对于某些$ k \ geq geq 1 $。我们还提供了用于保形维度的应用，并提供了一些以前已知的非安装结果的新证明。
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.
