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A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem

  • Simon Blatt
  • Manuel Schlierf
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Abstract

We study the Willmore flow with Dirichlet boundary conditions by means of a new geometric localization approach. In contrast to previous works, we do not rely on localized energy estimates, but on uniformly controlled isothermal coordinates and localized parabolic a priori estimates. For genus-zero surfaces satisfying an explicit energy condition depending only on the prescribed boundary data, we prove global existence together with a dichotomy between smooth convergence and subconvergence to a complete noncompact Willmore surface. If the initial Willmore energy is at most 4π, this dichotomy reduces to smooth convergence after reparametrization, and we show that this additional threshold is optimal. As an application, we obtain an existence criterion for the stationary Dirichlet Willmore problem.

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Cite

BibTeXRIS
BibTeX
@misc{blatt2026localization,
  title = {{A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem}},
  author = {Simon Blatt and Manuel Schlierf},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26602},
  eprint = {2610.26602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26602},
}
RIS
TY  - UNPB
TI  - A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem
AU  - Simon Blatt
AU  - Manuel Schlierf
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26602
UR  - https://arxiv.org/abs/2610.26602
AB  - We study the Willmore flow with Dirichlet boundary conditions by means of a new geometric localization approach. In contrast to previous works, we do not rely on localized energy estimates, but on uniformly controlled isothermal coordinates and localized parabolic a priori estimates. For genus-zero surfaces satisfying an explicit energy condition depending only on the prescribed boundary data, we prove global existence together with a dichotomy between smooth convergence and subconvergence to a complete noncompact Willmore surface. If the initial Willmore energy is at most 4π, this dichotomy reduces to smooth convergence after reparametrization, and we show that this additional threshold is optimal. As an application, we obtain an existence criterion for the stationary Dirichlet Willmore problem.
ER  -