Scale-critical curve diffusion flows
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914439501971456 |
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| author | Miura, Tatsuya Wheeler, Glen |
| author_facet | Miura, Tatsuya Wheeler, Glen |
| contents | We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates''. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $ω$-fold circle monotonically approaches the unit $ω$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $ω$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $ω$-circles depends non-monotonically on $ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01716 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scale-critical curve diffusion flows Miura, Tatsuya Wheeler, Glen Analysis of PDEs Differential Geometry 53E40, 35B35, 35B40, 35K55 We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates''. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $ω$-fold circle monotonically approaches the unit $ω$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $ω$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $ω$-circles depends non-monotonically on $ω$. |
| title | Scale-critical curve diffusion flows |
| topic | Analysis of PDEs Differential Geometry 53E40, 35B35, 35B40, 35K55 |
| url | https://arxiv.org/abs/2604.01716 |