Classification of zero-Euler-number minimal surfaces

Prove that the only minimal surfaces in \(\mathbb{S}^3\) bounded by a great circle with Euler number zero are hemispheres.

Background

The paper treats nonzero Euler number through the degree of the canonical family and obtains nontrivial min–max widths. It separately identifies the zero-Euler-number case as an unresolved classification problem, asking whether hemispheres are the only possibility.

References

A great circle in \mathbb{S}3 bounds no non-orientable embedded minimal surface with Euler number 0.

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Conjecture \ref{0}

The only minimal surfaces in $\mathbb{S}3$ bounded by $C$ with Euler number zero are the hemispheres.

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Question 8.3, Section 8 (Problems)

The following however is still open: \begin{question} Does $C$ bound any immersed orientable minimal surface aside from the hemispheres. \end{question}

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Question 8.4, Section 8 (Problems)