On the convergence of the Willmore flow with Dirichlet boundary conditions

Manuel Schlierf

doi:10.1016/j.na.2023.113475

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arXiv · 2303.05374

On the convergence of the Willmore flow with Dirichlet boundary conditions

Manuel Schlierf

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Abstract

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.

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Manuel Schlierf. 2023-03-09. On the convergence of the Willmore flow with Dirichlet boundary conditions. https://doi.org/10.1016/j.na.2023.113475

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