Year: 2026
Type: article
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.
Source: Forum of Mathematics Sigma
Authors: Elena Mäder-Baumdicker, Jona Seidel
Institutions: Technische Universität Darmstadt, Freie Universität Berlin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,650
OpenAlex ID: W4416513701