Uniqueness of minimizing leaves up to dilation

Determine whether the smooth properly embedded minimizing leaves associated with a general area-minimizing hypercone are unique up to dilation on each side of the cone.

Background

For a general area-minimizing hypercone, Zhihan Wang's theorem provides a smooth properly embedded minimizing leaf on each side of the cone, normalized by having distance one from the origin. Dilations of such a leaf generate a foliation of the corresponding side. Unlike the regular-cone setting, the paper uses only the existence of these leaves and explicitly records that uniqueness up to dilation is not known. Establishing uniqueness would clarify whether the resulting foliations, and potentially the associated geometric constructions, are canonical.

References

By the Wang's theorem Theorem~1.1, there is a smooth properly embedded minimizing leaf $\Sigma_\pm$ on each side $U_\pm$ with $dist(0,\Sigma_\pm)=1$ (different from , we only know the existence, while the uniqueness is unknown, so we can just choose one such foliation).

— A quantitative inequality for general area-minimizing hypercones  (2609.24958 - Niu, 21 Sep 2026) in Section 2, immediately after the statement of Theorem 2.1