Michael–Simon inequality for all atomic-condition anisotropies

Establish that every positive anisotropy satisfying the atomic condition (AC) supports a Michael–Simon inequality for varifolds with finite mass, finite anisotropic first variation, and a positive lower bound on the density.

Background

The paper assumes that the anisotropy supports a Michael–Simon inequality and combines this assumption with the quadratic exposed condition to prove an anisotropic Allard regularity theorem. The Michael–Simon inequality supplies the lower mass estimates needed in place of the monotonicity formula, which generally fails for anisotropic energies.

The authors note that existing results establish the inequality for several classes of anisotropies, including suitable perturbations of the area integrand, certain convex hypersurface anisotropies, and certain $\ell^q$ anisotropies. They explicitly expect the inequality to hold for the broader class of anisotropies satisfying the atomic condition, leaving this general statement unresolved. The paper proves the inequality for the specific axisymmetric families constructed later, but not for every AC anisotropy.