On the convergence of the Willmore flow with Dirichlet boundary conditions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929478889897984 |
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| author | Schlierf, Manuel |
| author_facet | Schlierf, Manuel |
| contents | Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions - it can even be made to be $0$. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li-Yau inequality for open curves in $\mathbb{H}^2$ is proved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_05374 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the convergence of the Willmore flow with Dirichlet boundary conditions Schlierf, Manuel Analysis of PDEs Differential Geometry 53E40 (Primary), 35B40, 35K41 (Secondary) Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions - it can even be made to be $0$. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li-Yau inequality for open curves in $\mathbb{H}^2$ is proved. |
| title | On the convergence of the Willmore flow with Dirichlet boundary conditions |
| topic | Analysis of PDEs Differential Geometry 53E40 (Primary), 35B40, 35K41 (Secondary) |
| url | https://arxiv.org/abs/2303.05374 |