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On the Willmore energy of flat n-tori in โ„N and Chen's conjecture for n-tori

  • Ruijie Ni
  • Peng Wang
  • Zhenxiao Xie
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Abstract

This paper establishes the sharp lower bound (4nฯ€2)n/2 for the Willmore energy ๐’ฒ of flat n-tori in the Euclidean space. Up to Mรถbius transformations, the Clifford n-torus ๐•Š1(โˆš(1/n) ) ร— โ‹ฏ ร— ๐•Š1(โˆš(1/n) ) โŠ‚ ๐•Š2nโˆ’1 โŠ‚ โ„2n is shown to be the unique minimizer attaining this bound. This also confirms Chen's conjecture for flat n-tori. However, when $n \geq3 $, we show that Chen's conjecture fails on the total mean curvature of general immersed n-tori: certain Mรถbius transformations of the Clifford n-torus strictly decrease the total mean curvature.

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Cite

BibTeXRIS
BibTeX
@misc{ni2026willmore,
  title = {{On the Willmore energy of flat $n$-tori in $\mathbbR^N$ and Chen's conjecture for $n$-tori}},
  author = {Ruijie Ni and Peng Wang and Zhenxiao Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36491},
  eprint = {2609.36491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36491},
}
RIS
TY  - UNPB
TI  - On the Willmore energy of flat n-tori in โ„^N and Chen's conjecture for n-tori
AU  - Ruijie Ni
AU  - Peng Wang
AU  - Zhenxiao Xie
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36491
UR  - https://arxiv.org/abs/2609.36491
AB  - This paper establishes the sharp lower bound (4nฯ€^2)^n/2 for the Willmore energy ๐’ฒ of flat n-tori in the Euclidean space. Up to Mรถbius transformations, the Clifford n-torus ๐•Š^1(โˆš(1/n) ) ร— โ‹ฏ ร— ๐•Š^1(โˆš(1/n) ) โŠ‚ ๐•Š^2nโˆ’1 โŠ‚ โ„^2n is shown to be the unique minimizer attaining this bound. This also confirms Chen's conjecture for flat n-tori. However, when $n \geq3 $, we show that Chen's conjecture fails on the total mean curvature of general immersed n-tori: certain Mรถbius transformations of the Clifford n-torus strictly decrease the total mean curvature.
ER  -