Second-lowest-area minimal surface spanning a great circle

Establish that the minimal surface of second-lowest area in the round three-sphere bounded by a great circle is the Lawson Möbius band \(\overline{\tau}_{1,2}\).

Background

The paper identifies hemispheres as the area-minimizing orientable minimal surfaces spanning a great circle and proposes that the embedded Lawson Möbius band τ‾1,2\overline{\tau}_{1,2} is the next simplest and has the second-lowest area. This conjecture is presented as the boundary analogue of the Marques–Neves characterization of the Clifford torus as the closed minimal surface of second-lowest area.

References

While the hemispheres are the simplest minimal surfaces spanning $C$, we expect the Lawson band, $\bar{\tau}_{1,2}$ to be the second simplest:

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Conjecture 1.1 (Introduction)

we expect, by analogy with F. Urbano's theorem , this to characterize $\bar{\tau}_{1,2}$:

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Conjecture 1.2 (Introduction)

If $\Sigma$ is a $C2$ immersion of a Möbius band into $\mathbb{S}3$ that spans $C$, then $$ \mathcal{W}(\Sigma)\geq \mathcal{W}(\bar{\tau}{1,2}) = \mathrm{Area}(\bar{\tau}{1,2}).$$

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Conjecture 1.4 (Introduction)

The area of a non-orientable minimal surface in $\mathbb{S}3$ spanning $C$ with non-zero Euler number is bounded from below by that of an embedded non-orientable minimal surface with index $2$ and boundary $C$.

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Question 8.2, Section 8 (Problems)