The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909665731805184 |
|---|---|
| author | Mäder-Baumdicker, Elena Seidel, Jona |
| author_facet | Mäder-Baumdicker, Elena Seidel, Jona |
| contents | We study the sublevel sets of the Willmore energy on the space of smoothly immersed $ 2 $-spheres in Euclidean $ 3 $-space. We show that the subset of immersions with energy at most $ 12π$ 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 $ 12π$ that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li-Yau inequality at $ 12π$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow Mäder-Baumdicker, Elena Seidel, Jona Differential Geometry 53C42, 53E40, 57R42 We study the sublevel sets of the Willmore energy on the space of smoothly immersed $ 2 $-spheres in Euclidean $ 3 $-space. We show that the subset of immersions with energy at most $ 12π$ 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 $ 12π$ that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li-Yau inequality at $ 12π$ 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. |
| title | The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow |
| topic | Differential Geometry 53C42, 53E40, 57R42 |
| url | https://arxiv.org/abs/2506.23359 |