Prescribed Genus in the Round Four-Sphere

Determine which integers $\gamma\geq2$ occur as the genus of a smooth closed connected orientable embedded minimal surface in the unit round four-sphere with Gaussian curvature strictly negative everywhere, and prove or disprove that every sufficiently large integer occurs.

Background

The construction produces embedded minimal surfaces in S4\mathbb S^4 of genus m2+1m^2+1 for every sufficiently large integer mm. Consequently, it realizes only a sparse prescribed sequence of genera rather than classifying all possible genera.

The lower bound γ≥2\gamma\geq2 follows from Gauss–Bonnet, while the smallest possible genus and the realization of genera outside the form m2+1m^2+1 remain unresolved.

References

Which integers $\gamma\geq2$ occur as the genus of a smooth closed connected orientable embedded minimal surface in $\mathbb{S}4$ with everywhere negative Gaussian curvature? In particular, does every sufficiently large integer occur?

— Negatively Curved Minimal Surfaces in the Round Four-Sphere  (2610.12238 - Caniato, 8 Oct 2026) in Question 2, Section 1, subsection “Open problems”