Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations

What the math gives to ML

The paper provides an explicit anisotropic probing identity for the minimal-hypersurface operator: after diagonal dilation, minimality becomes a finite generalized power sum whose coefficients are indexed by coordinate-pair weight sums. Under distinct pair sums, evaluating the operator at finitely many dilation scales makes the coefficient map invertible, so cancellation at multiple scales is much stronger than cancellation at one scale. This can transfer to implicit neural representations as a structured curvature regularizer or diagnostic that separately penalizes coordinate-pair second-order interactions rather than only their aggregate contraction. The likely use is not generic network training, but learning level sets that are locally planar, anisotropically simple, or robust under prescribed coordinate rescalings.