Minimal Willmore energy by topological type

Determine which topological type, among closed submanifolds other than the sphere, realizes the least Willmore energy in higher-dimensional submanifold theory.

Background

The paper compares the flat-torus lower bound with the Guo–Li–Wang conjectural lower bound for submanifolds homeomorphic to products of spheres. This comparison raises the broader unresolved issue of identifying the topology that minimizes Willmore energy once the spherical topology is excluded.

References

In higher-dimensional submanifold theory, however, it is not known which topological type should realize the least Willmore energy among those different from $\mathbb{S}n$.

— On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori  (2609.36491 - Ni et al., 29 Sep 2026) in Remark following Corollary \ref{thm-mini-iso}, Section 3