Introducing spontaneous curvature to the Helfrich flow: Singularities and convergence
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| Format: | Preprint |
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2024
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| _version_ | 1866917645184401408 |
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| author | Schlierf, Manuel |
| author_facet | Schlierf, Manuel |
| contents | While there are various results on the long-time behavior of the Willmore flow, the Helfrich flow with non-zero spontaneous curvature as its natural generalization is not yet well-understood. Past results for the gradient flow of a locally area- and volume-constrained Willmore flow indicate the existence of finite-time singularities which corresponds to the scaling-behavior of the underlying energy. However, for a non-vanishing spontaneous curvature, the scaling behavior is not quite as conclusive.
Indeed, in this article, we find that a negative spontaneous curvature corresponds to finite-time singularities of the locally constrained Helfrich flow if the initial surface is energetically close to a round sphere. Conversely however, in the case of a positive spontaneous curvature, we find a positive result in terms of the convergence behavior: The locally area-constrained Helfrich flow starting in a spherical immersion with suitably small Helfrich energy exists globally and converges to a Helfrich immersion after reparametrization. Moreover, this energetic smallness assumption is given by an explicit energy threshold depending on the spontaneous curvature and the local area constraint of the energy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_12820 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Introducing spontaneous curvature to the Helfrich flow: Singularities and convergence Schlierf, Manuel Analysis of PDEs Differential Geometry 53E40, 49Q10 (primary), 35B40, 35K41 (secondary) While there are various results on the long-time behavior of the Willmore flow, the Helfrich flow with non-zero spontaneous curvature as its natural generalization is not yet well-understood. Past results for the gradient flow of a locally area- and volume-constrained Willmore flow indicate the existence of finite-time singularities which corresponds to the scaling-behavior of the underlying energy. However, for a non-vanishing spontaneous curvature, the scaling behavior is not quite as conclusive. Indeed, in this article, we find that a negative spontaneous curvature corresponds to finite-time singularities of the locally constrained Helfrich flow if the initial surface is energetically close to a round sphere. Conversely however, in the case of a positive spontaneous curvature, we find a positive result in terms of the convergence behavior: The locally area-constrained Helfrich flow starting in a spherical immersion with suitably small Helfrich energy exists globally and converges to a Helfrich immersion after reparametrization. Moreover, this energetic smallness assumption is given by an explicit energy threshold depending on the spontaneous curvature and the local area constraint of the energy. |
| title | Introducing spontaneous curvature to the Helfrich flow: Singularities and convergence |
| topic | Analysis of PDEs Differential Geometry 53E40, 49Q10 (primary), 35B40, 35K41 (secondary) |
| url | https://arxiv.org/abs/2404.12820 |