Quasiregular curves: Removability of singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916531892387840 |
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| author | Ikonen, Toni |
| author_facet | Ikonen, Toni |
| contents | We prove a Painlevé theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves.
We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and an injectivity radius lower bound. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $ω$-curve from $\mathbb{R}^n$ into $( N, ω)$, where the bounded cohomology class of $ω$ is in the bounded Künneth ideal, has infinite energy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_02334 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasiregular curves: Removability of singularities Ikonen, Toni Differential Geometry Complex Variables Primary: 30C65. Secondary: 49Q15, 53C65 We prove a Painlevé theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves. We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and an injectivity radius lower bound. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $ω$-curve from $\mathbb{R}^n$ into $( N, ω)$, where the bounded cohomology class of $ω$ is in the bounded Künneth ideal, has infinite energy. |
| title | Quasiregular curves: Removability of singularities |
| topic | Differential Geometry Complex Variables Primary: 30C65. Secondary: 49Q15, 53C65 |
| url | https://arxiv.org/abs/2407.02334 |