Critical catenoid conjecture

Every smooth properly embedded minimal surface $\Sigma\subset \overline{\mathbb{B}}^{3}$ that is topologically an annulus, satisfies $\partial\Sigma\subset \mathbb{S}^{2}$ , and meets $\mathbb{S}^{2}=\partial\mathbb{B}^{3}$ orthogonally along $\partial\Sigma$ is congruent to the critical catenoid; equivalently, there exists an orthogonal transformation $Q\in O(3)$ such that $\Sigma=Q(\mathcal{C}_{\mathrm{crit}})$ , where $\mathcal{C}_{\mathrm{crit}}$ is the critical catenoid in the unit ball.

References

Primary source

arXiv

Additional references

Uniqueness of free boundary minimal annuli — arXiv — Davide Parise, Jonathan J. Zhu

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture, but no independent verification has appeared.

The conjecture says that every embedded free-boundary minimal annulus in the three-dimensional unit ball is congruent to the critical catenoid. It was formulated by Nitsche in 1985 and later identified as the Fraser–Li conjecture.

Known results

Fraser and Schoen: uniqueness under the condition $\sigma_{1}=1$ .
Tran (2020): uniqueness among annuli of Morse index $4$ .
Espinar and Marín (2023): uniqueness under an additional support-function hypothesis.
Non-rotational immersed examples exist, but they are not embedded and therefore do not refute the conjecture.

September 2026 claimed solution

Davide Parise and Jonathan J. Zhu’s preprint claims uniqueness of the free-boundary minimal annulus and classification of the associated annular spherical Bernoulli solutions, which would settle the conjecture. The claim is unrefereed and has not been independently corroborated in the retrieved material.

Current status (as of September 2026): Earlier partial results are established, while the full conjecture is only claimed solved by the new preprint and remains unverified.

Sources

No solutions have been posted yet.