Persistence of spherical suspensions of non-orientable minimal surfaces

Determine whether the spherical suspension in the round sphere of a minimal non-orientable surface of any positive non-orientable genus, such as a Klein bottle, persists under sufficiently small perturbations of the ambient metric in the sense of Theorem 1.1, namely, whether it remains an observed minimal submanifold with strongly isolated singularities on an open set of metrics.

Background

The paper establishes persistence results for certain minimal submanifolds with strongly isolated singularities (MSI) under perturbations of the ambient Riemannian metric. In particular, the authors show that spherical suspensions of certain closed minimal submanifolds in round spheres are persistent when the effective index and effective nullity vanish and the relevant Jacobi fields are integrable.

The authors further note that a result of Bernstein and Ketover rules out persistence for spherical suspensions of closed orientable minimal surfaces in the relevant setting. The unresolved case concerns minimal non-orientable surfaces of positive non-orientable genus, including the Klein bottle, for which the paper does not determine whether the corresponding spherical suspensions exhibit persistence.