Classification of nonsmooth minimizing tangent maps in complex projective space

Classify all minimizing tangent maps into \(\mathbb{CP}^n\) whose links are not smooth, and determine whether, at least for \(n=1\), the cylindrical extensions of the minimizing tangent maps classified in Theorem 3d constitute the only additional family beyond the smooth-link maps classified in Theorem 4d.

Background

Theorem 4d classifies minimizing tangent maps $\mathbb{B}^4\to\mathbb{CP}^n$ under the assumption that the link $u:\mathbb{S}^3\to\mathbb{CP}^n$ is smooth. The paper notes that this smoothness assumption cannot simply be removed because cylindrical extensions of the minimizing maps from the three-dimensional classification provide additional minimizing tangent maps with non-isolated singularities.

The authors conjecture that, at least when $n=1$, these cylindrical examples are the only other family. The obstruction is that the argument proving factorization through the Hopf map relies on smoothness and cannot presently be extended to links with singularities.