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Peer-reviewedCC BY 4.0Forum of Mathematics, Sigma

The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow

  • Elena Mäder-Baumdicker1,2
  • Jona Seidel2
  1. 1Freie Universität Berlin
  2. 2Technical University Darmstadt
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Abstract

We study the sublevel sets of the Willmore energy on the space of smoothly immersed 2 $ 2 $ 2 -spheres in Euclidean 3 $ 3 $ 3 -space. We show that the subset of immersions with energy at most 12 π $ 12\pi $ 12 pi consists of four regular homotopy classes. Moreover, we show that, in certain regular homotopy classes, all singularities of the Willmore flow are avoidable – that is, the initial surface admits a regular homotopy to a round sphere whose Willmore energy does not exceed that of the initial surface. This yields a classification of initial surfaces with energy at most 12 π $ 12\pi $ 12 pi that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li–Yau inequality at 12 π $ 12\pi $ 12 pi for a large class of immersed spheres without triple points. To prove these results, we glue together different instances of the Willmore flow and employ an invariant for triple-point-free immersed spheres.

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Cite

BibTeXRIS
BibTeX
@article{maderbaumdicker2026willmore,
  title = {{The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow}},
  author = {Elena Mäder-Baumdicker and Jona Seidel},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10282},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102825/type/journal_article},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
AU  - Elena Mäder-Baumdicker
AU  - Jona Seidel
PY  - 2026
DA  - 2026/10/08
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10282
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102825/type/journal_article
AB  - We study the sublevel sets of the Willmore energy on the space of smoothly immersed 2 $ 2 $ 2 -spheres in Euclidean 3 $ 3 $ 3 -space. We show that the subset of immersions with energy at most 12 π $ 12\pi $ 12 pi consists of four regular homotopy classes. Moreover, we show that, in certain regular homotopy classes, all singularities of the Willmore flow are avoidable – that is, the initial surface admits a regular homotopy to a round sphere whose Willmore energy does not exceed that of the initial surface. This yields a classification of initial surfaces with energy at most 12 π $ 12\pi $ 12 pi that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li–Yau inequality at 12 π $ 12\pi $ 12 pi for a large class of immersed spheres without triple points. To prove these results, we glue together different instances of the Willmore flow and employ an invariant for triple-point-free immersed spheres.
SN  - 2050-5094
ER  -