Linearly Full Embedded Superminimal Surfaces in the Six-Sphere

Construct, or prove impossible, a smooth closed connected embedded linearly full superminimal surface in $\mathbb S^6$ with Gaussian curvature strictly negative everywhere.

Background

Superminimal surfaces form a more restrictive class of minimal surfaces characterized, in the four-sphere case, by a circular image of the unit tangent circle under the second fundamental form and, in higher even-dimensional spheres, by corresponding twistor-lift conditions.

The paper’s negatively curved examples in higher-dimensional spheres arise through totally geodesic inclusions of $\mathbb S^4$ and therefore are not linearly full in $\mathbb S^6$. The paper consequently leaves open whether an unbranched embedded example exists that is genuinely linearly full and superminimal in the six-sphere.