Uniqueness of the embedded minimal Möbius band

Prove that every embedded minimal Möbius band in \(\mathbb{S}^3\) spanning a great circle is, up to ambient isometry, the Lawson Möbius band \(\overline{\tau}_{1,2}\).

Background

The paper proves that an area-minimizing minimal Möbius band is embedded, has index two, and has Euler number ±2\pm2, but does not establish uniqueness. The conjecture would identify all embedded minimal Möbius bands spanning the prescribed great circle with the Lawson example.

References

The following is a natural analog to the Lawson Conjecture (resolved by S. Brendle ): The Lawson Möbius band is the unique embedded minimal Möbius band with boundary a great circle.

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Questions

The Lawson Möbius band is the unique embedded non-orientable minimal surface with boundary a great circle and with Euler number \pm 2.

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Conjecture \ref{2}

S. Brendle's resolution of the Lawson conjecture also suggests: \begin{conjecture}\label{L2Conj} An embedded minimal Möbius band in $\mathbb{S}3$ spanning $C$ and is (up to ambient isometry) the Lawson band $\overline{\tau}_{1,2}$. \end{conjecture}

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Conjecture 1.3 (Introduction)

Can one use the mean curvature flow in $\mathbb{S}3$ to prove Theorems \ref{mainintro} and \ref{ConfAreaLBThm}?

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Question, Section 8 (Problems)

More generally, one can ask whether the lower bound in Theorem \ref{admissible} is saturated: There exists no embedded minimal surface (aside from the Lawson Möbius band) with boundary a great circle and genus equal to half of its Euler number.

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Questions