Montiel–Urbano Willmore-type conjecture

Let $T_{\mathrm{Cl}}\subset\mathbb{C}P^2$ denote the Clifford torus, and for a closed immersed surface $\Sigma\subset\mathbb{C}P^2$ define $\mathcal{W}^{-}(\Sigma)=\int_{\Sigma}(2+\lvert H\rvert^2)\,dA$ , where $H$ is the mean-curvature vector. The Montiel–Urbano conjecture asks whether $\mathcal{W}^{-}(\Sigma)\geq \mathcal{W}^{-}(T_{\mathrm{Cl}})$ for every torus $\Sigma\subset\mathbb{C}P^2$ , or at least for every Lagrangian torus $\Sigma\subset\mathbb{C}P^2$ .

References

Primary source

arXiv

Additional references

A proof of the Willmore-type conjecture in CP² — arXiv — Peng Wang, Zhenxiao Xie, Chen Zhao

differential-geometry geometric-analysis

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new paper claims the conjecture holds for constrained surfaces but fails without that constraint; the claims have not been independently verified.

Montiel and Urbano conjectured that the Clifford torus is extremal for a Willmore-type energy among all tori in $\mathbb{C}P^{2}$ , or among Lagrangian tori. The new work separates these two formulations.

Known results

Wang and Xie (2023) established strong Willmore stability of the Clifford torus, supporting—but not proving—the conjecture tracked here.

September 2026 claimed resolution

Peng Wang, Zhenxiao Xie, and Chen Zhao claim to prove the Lagrangian version, with equality only for the Clifford torus, and to disprove the unrestricted version via non-Lagrangian deformations that lower the energy. The arXiv source also discloses that ChatGPT assisted with verification, proof strategy, counterexamples, and exposition; the authors state that they checked and rewrote the mathematics. These results remain unverified.

Current status (as of September 2026): The Lagrangian version is claimed proved and the unrestricted version claimed disproved, but both claims remain unverified.

Sources

No solutions have been posted yet.