For every integer n≥ 2, every integer N≥ n+1, and every immersion f:T^n→R^N whose induced metric is flat, the Willmore energy W(f)=int_T^n| H|^n,d mu satisfies W(f)≥ (4n pi^2)^n/2. Equality holds precisely, up to Möbius transformations, for the Clifford n-torus S^1(√1/n)×…×S^1(√1/n)⊂S^2n-1⊂R^2n.

Chen’s conjecture for flat n-tori

For every integer $n\geq 2$ , every integer $N\geq n+1$ , and every immersion $f:T^n\to\mathbb{R}^N$ whose induced metric is flat, the Willmore energy $\mathcal{W}(f)=\int_{T^n}\lvert H\rvert^n\,d\mu$ satisfies $\mathcal{W}(f)\geq (4n\pi^2)^{n/2}$ . Equality holds precisely, up to Möbius transformations, for the Clifford $n$ -torus $\mathbb{S}^1(\sqrt{1/n})\times\cdots\times\mathbb{S}^1(\sqrt{1/n})\subset\mathbb{S}^{2n-1}\subset\mathbb{R}^{2n}$ .

References

Primary source

arXiv

Additional references

On the Willmore energy of flat n-tori in R^N and Chen's conjecture for n-tori — arXiv — Ruijie Ni, Peng Wang, Zhenxiao Xie

differential-geometry willmore-geometry

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

An unrefereed preprint claims to settle the conjecture for flat tori, while showing that its broader version fails for more general tori.

Chen’s conjecture predicts a sharp bound and equality classification for flat $n$ -tori. The reported result is restricted to the flat setting and separates it from a broader, false formulation.

Recent preprint

Ruijie Ni, Peng Wang, and Zhenxiao Xie claim the sharp bound $(4n\pi^2)^{n/2}$ for flat $n$ -tori, including the equality classification. They also state that the corresponding total-mean-curvature assertion fails for general immersed $n$ -tori when $n \ge 3$ . This is an unrefereed preprint, so the resolution is reported as unverified.

Current status (as of September 2026): The flat $n$ -torus formulation is claimed solved by an unrefereed preprint, while the broader formulation is reported false for $n \ge 3$ ; independent verification is not recorded.

Sources

No solutions have been posted yet.