Coincidence of the two high-dimensional invariant solution families

Establish whether, for sufficiently small positive scaling parameter, the O(m)×O(n)-invariant Allen–Cahn solution families constructed from the Lawson-cone hypersurfaces [?]Sigma^7_{m,n} and the solution families obtained in Theorem 1.2 of Pacard and Wei coincide, in the dimensions where the corresponding Lawson cones are minimizing.

Background

When m+n≥8 with m,n≥3, or when m+n≥9, the Lawson cone C_{m,n} is minimizing. In that setting, the minimal hypersurfaces used in the paper’s construction coincide with the hypersurfaces appearing in the construction of Pacard and Wei, so both approaches produce O(m)×O(n)-invariant Allen–Cahn solution families.

Although the geometric input is the same in these cases, the paper does not establish that the resulting solution families are identical. The conjecture concerns equality for sufficiently small positive scaling parameter, rather than merely qualitative agreement or asymptotic similarity.