Curve-flat functions and Lipschitz quotients
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915877735104512 |
|---|---|
| author | Kaasik, Jaan Kristjan Quilis, Andrés |
| author_facet | Kaasik, Jaan Kristjan Quilis, Andrés |
| contents | We show that for every complete metric space $M$ there exists another complete metric space $N$ of the same density character such that the curve-flat quotient of $N$ is isometric to $M$. Moreover, we show that if $M$ is compact and $α$ is any countable ordinal, there exists a compact $N$ such that its curve-flat quotient of order $α$ is bi-Lipschitz equivalent to $M$, with arbitrarily small distortion. Our constructions rely on a new method for constructing (compact) metric spaces, which consists in attaching iteratively compact spaces at countably many pairs of points to a snowflake-like distortion of a given (compact) metric space.
We apply our results on high-order curve-flat quotients to obtain a new result concerning universality of Lipschitz quotients. Specifically, we show that there cannot exist a compact metric space $K$ such that every compact metric space is a Lipschitz quotient of $K$. This result stands in contrast to a theorem of Johnson, Lindenstrauss, Preiss and Schechtman, who showed that any separable Banach space containing $\ell_1$ has every separable geodesic complete metric space as a Lipschitz quotient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20177 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curve-flat functions and Lipschitz quotients Kaasik, Jaan Kristjan Quilis, Andrés Metric Geometry Functional Analysis 51F30 (Primary) 46B80 (Secondary) We show that for every complete metric space $M$ there exists another complete metric space $N$ of the same density character such that the curve-flat quotient of $N$ is isometric to $M$. Moreover, we show that if $M$ is compact and $α$ is any countable ordinal, there exists a compact $N$ such that its curve-flat quotient of order $α$ is bi-Lipschitz equivalent to $M$, with arbitrarily small distortion. Our constructions rely on a new method for constructing (compact) metric spaces, which consists in attaching iteratively compact spaces at countably many pairs of points to a snowflake-like distortion of a given (compact) metric space. We apply our results on high-order curve-flat quotients to obtain a new result concerning universality of Lipschitz quotients. Specifically, we show that there cannot exist a compact metric space $K$ such that every compact metric space is a Lipschitz quotient of $K$. This result stands in contrast to a theorem of Johnson, Lindenstrauss, Preiss and Schechtman, who showed that any separable Banach space containing $\ell_1$ has every separable geodesic complete metric space as a Lipschitz quotient. |
| title | Curve-flat functions and Lipschitz quotients |
| topic | Metric Geometry Functional Analysis 51F30 (Primary) 46B80 (Secondary) |
| url | https://arxiv.org/abs/2603.20177 |