Quadratic Flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Let Ω be an open set in a Euclidean space X of dimension (n+1) and ϕ be a uniformly convex smooth norm on X. Consider an n-dimensional unit-density varifold V in Ω, whose generalised mean curvature vector, computed with respect to ϕ, is bounded. Assume also that the n-dimensional Hausdorff measure restricted to the support Σ of V is absolutely continuous with respect to the weight measure of V. In my recent work with Mario Santilli (arXiv:2507.18357), we showed that there exists an open and dense subset R of Σ at points of which one can touch Σ by two mutually tangent balls. At points of R we get quadratic height decay and we then apply Allard's 1986 regularity theorem to show that these points are actually regular points of class (1,α) for any 0 < α < 1. We prove also that R is almost equal to a subset Σ* of points of Σ, where at least one blow-up limit (in the sense of Painlevé-Kuratowski) is not the whole space X. The condition on the blow-up limit seems very weak and does not entail any regularity a priori; hence, there is hope that in many cases Σ* almost equals Σ. In my talk I shall outline the history of the problem of regularity for varifolds satisfying bounds on anisotropic first variation and I shall present main ingredients of the proof of the above result.