Convergence to saddle-shaped solutions in the equal-factor Lawson-cone case

Determine whether, when m=n, the O(m)×O(n)-invariant Allen–Cahn solutions constructed from the hypersurfaces [?]Sigma^7_{m,n} converge in L^(R^{N+1}) to the saddle-shaped solutions constructed by Cabre9 and Terra, with convergence in a finer topology away from the singularity of the cone.

Background

For m=n, the Lawson cone C_{m,m} is the Simon cone, and Corollary 3.1 constructs O(m)×O(n)-invariant stable Allen–Cahn solutions whose nodal sets are asymptotic to minimal hypersurfaces associated with this cone. The paper compares these solutions with the saddle-shaped solutions previously constructed by Cabre9 and Terra, which possess the same symmetry.

The unresolved issue is whether the two constructions yield the same limiting solution as the scaling parameter tends to zero. The authors further specify that the expected convergence should hold in the uniform norm on the whole space and in a stronger topology away from the cone’s singularity.

References

In the case $m=n$, we conjecture that the solutions $u_{}$ from Corollary \ref{cor-cz-highdim} converge in $L{\infty}(R{N+1})$ to the solutions presented in , with convergence in a finer topology away from the singularity of the cone.

— The Allen-Cahn equation and general minimal cones  (2610.01335 - Rico et al., 1 Oct 2026) in Section 3, subsection “O(m)×O(n)-invariant solutions in dimension m+n≥8”