Tangents to Lipschitz and Sobolev images
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918314562813952 |
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| author | Badger, Matthew Krandel, Jared Vellis, Vyron |
| author_facet | Badger, Matthew Krandel, Jared Vellis, Vyron |
| contents | We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from $[0,1]^n$ into a Euclidean space with $p>n$, the image $f([0,1]^n)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from $[0,1]^n$ into a metric space, we show that the image $f([0,1]^n)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in $\mathbb{R}^d$. We show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the $n$-packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22473 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tangents to Lipschitz and Sobolev images Badger, Matthew Krandel, Jared Vellis, Vyron Metric Geometry Classical Analysis and ODEs Primary 28A75, Secondary 46E36, 53C23 We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from $[0,1]^n$ into a Euclidean space with $p>n$, the image $f([0,1]^n)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from $[0,1]^n$ into a metric space, we show that the image $f([0,1]^n)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in $\mathbb{R}^d$. We show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the $n$-packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content. |
| title | Tangents to Lipschitz and Sobolev images |
| topic | Metric Geometry Classical Analysis and ODEs Primary 28A75, Secondary 46E36, 53C23 |
| url | https://arxiv.org/abs/2601.22473 |