Sharp Willmore lower bounds beyond the torus setting

Establish sharp lower bounds for the Willmore energy of submanifolds in higher codimension and for surfaces of higher genus.

Background

The paper places its result within the broader problem of determining sharp lower bounds for conformally invariant curvature functionals. It notes that, despite substantial progress for embedded two-dimensional tori in low codimension, sharp estimates remain unresolved in other geometric settings, specifically higher codimension and higher genus.

References

Nevertheless, sharp lower-bound problems for the Willmore energy remain open in several settings, including higher codimension and higher genus surfaces.

— 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 Section 1, Introduction

Guo-Li-Wang proposed the following conjecture in . \begin{conjecture}[{\rm Guo-Li-Wang}] Let $M_k$ be an $n$-dimensional closed manifold homeomorphic to $Sk\times S{n-k}$. If $f:M_k\to S{n+1}$ is an embedding, then $$\mathcal W(f)\ge B_{n,k}\triangleq\frac{4\pi{\frac{n+2}{2}(n-k){\frac{k}{2}k{\frac{n-k}{2}{n{\frac n2-2}(n-1)\Gamma\left(\frac{k+1}{2}\right)\Gamma\left(\frac{n-k+1}{2}\right)}.$$ \end{conjecture}

— 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 Conjecture in the remark following Corollary \ref{thm-mini-iso}, Section 3