Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917671340081152 |
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| author | Khan, Gabriel |
| author_facet | Khan, Gabriel |
| contents | We study curve-shortening flow for twisted curves in $\mathbb{R}^3$ (i.e., curves with nowhere vanishing curvature $κ$ and torsion $τ$) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II singularity which admits sequences along which $\fracτ{κ^2} \to \infty$. This contrasts strongly with Altschuler's planarity theorem [J. Differential Geom. (1991)], which shows that along any essential blow-up sequence, $\fracτκ \to 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07171 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow Khan, Gabriel Differential Geometry Analysis of PDEs 53E10 53A04 We study curve-shortening flow for twisted curves in $\mathbb{R}^3$ (i.e., curves with nowhere vanishing curvature $κ$ and torsion $τ$) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II singularity which admits sequences along which $\fracτ{κ^2} \to \infty$. This contrasts strongly with Altschuler's planarity theorem [J. Differential Geom. (1991)], which shows that along any essential blow-up sequence, $\fracτκ \to 0$. |
| title | Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow |
| topic | Differential Geometry Analysis of PDEs 53E10 53A04 |
| url | https://arxiv.org/abs/2305.07171 |