Rigidity under disjoint four-index resonance

Determine which grouped coefficient systems arising from pairwise distinct dilation weights with at least one four-index resonance between disjoint pairs still force the second fundamental form to vanish in higher dimensions or when several simultaneous four-index resonances occur.

Background

The paper proves affine rigidity for connected embedded Euclidean minimal hypersurfaces whose images under finitely many members of a positive diagonal dilation family remain minimal, provided all unordered pair sums of the dilation weights are distinct. When pair sums coincide, the dilation identity determines only grouped sums of the pair coefficients, allowing curvature cancellation that coefficient separation cannot exclude.

The paper settles shared-index resonances by observing that they force repeated weights and by constructing nonflat helicoidal examples in every dimension. It also gives a four-dimensional nonflat minimal quadratic cone for the disjoint resonance 1+4=2+3. What remains unresolved is the broader classification for pairwise distinct weights with disjoint four-index resonances, especially in higher dimensions and in systems containing multiple such resonances.