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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.29769 |
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| _version_ | 1866914435309764608 |
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| author | Esmayli, Behnam Koskela, Pekka Nguyen, Khanh |
| author_facet | Esmayli, Behnam Koskela, Pekka Nguyen, Khanh |
| contents | A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff--$(n-1)$ measure. In short, we say that we can allow an exceptional codimension $1$ subset.
In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension $p$ subset, $p \ge 1$, if the source space satisfies a $p$-Poincaré inequality.
We prove, effectively, the sharpness of the latter claim. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29769 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II Esmayli, Behnam Koskela, Pekka Nguyen, Khanh Functional Analysis Complex Variables Metric Geometry A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff--$(n-1)$ measure. In short, we say that we can allow an exceptional codimension $1$ subset. In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension $p$ subset, $p \ge 1$, if the source space satisfies a $p$-Poincaré inequality. We prove, effectively, the sharpness of the latter claim. |
| title | Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II |
| topic | Functional Analysis Complex Variables Metric Geometry |
| url | https://arxiv.org/abs/2603.29769 |