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PreprintarXiv licencearXiv

A proof of the Willmore-type conjecture in ℂP2

  • Peng Wang
  • Zhenxiao Xie
  • Chen Zhao
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Abstract

In 2002, Montiel and Urbano conjectured that the Clifford torus in ℂP2 minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus g≥ 1 in ℂP2 has 𝒲--energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which 𝒲- strictly decreases.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026proof,
  title = {{A proof of the Willmore-type conjecture in $\mathbbCP^2$}},
  author = {Peng Wang and Zhenxiao Xie and Chen Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26721},
  eprint = {2609.26721},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26721},
}
RIS
TY  - UNPB
TI  - A proof of the Willmore-type conjecture in ℂP^2
AU  - Peng Wang
AU  - Zhenxiao Xie
AU  - Chen Zhao
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26721
UR  - https://arxiv.org/abs/2609.26721
AB  - In 2002, Montiel and Urbano conjectured that the Clifford torus in ℂP^2 minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus g≥ 1 in ℂP^2 has 𝒲^--energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which 𝒲^- strictly decreases.
ER  -