Stability of the free boundary Willmore problem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911399397031936 |
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| author | Dall'Acqua, Anna Rupp, Fabian Schätzle, Reiner Schlierf, Manuel |
| author_facet | Dall'Acqua, Anna Rupp, Fabian Schätzle, Reiner Schlierf, Manuel |
| contents | We study the Willmore problem with free boundary by means of a new Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation of the Fréchet derivative, but merely on an inequality. For the free boundary Willmore flow, we prove that solutions starting sufficiently close to a local minimizer exist for all times and converge. In the static setting, we prove quantitative stability of free boundary Willmore immersions and a local rigidity result in a neighborhood of free boundary minimal surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18388 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of the free boundary Willmore problem Dall'Acqua, Anna Rupp, Fabian Schätzle, Reiner Schlierf, Manuel Analysis of PDEs Differential Geometry 53E40 (primary), 35R35, 58E12, 26D10 (secondary) We study the Willmore problem with free boundary by means of a new Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation of the Fréchet derivative, but merely on an inequality. For the free boundary Willmore flow, we prove that solutions starting sufficiently close to a local minimizer exist for all times and converge. In the static setting, we prove quantitative stability of free boundary Willmore immersions and a local rigidity result in a neighborhood of free boundary minimal surfaces. |
| title | Stability of the free boundary Willmore problem |
| topic | Analysis of PDEs Differential Geometry 53E40 (primary), 35R35, 58E12, 26D10 (secondary) |
| url | https://arxiv.org/abs/2601.18388 |