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arXiv · 2603.20177

Universality of Lipschitz quotients and the curve-flat index

Jaan Kristjan Kaasik·Andrés Quilis
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Abstract

We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $\alpha$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $\alpha$ is almost-isometric to $M$.

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Jaan Kristjan Kaasik, Andrés Quilis. 2026-03-20. Universality of Lipschitz quotients and the curve-flat index. https://arxiv.org/abs/2603.20177

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