Seminar in Geometric Function and Mapping Theory

Abstract

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 joint recent work with Mario Santilli (arXiv:2507.18357), we showed that at ℋⁿ alomost all points of Σ one can touch Σ by two mutually tangent balls. At these points we get quadratic height decay and we then apply Allard’s 1986 regularity theorem to show that thees points are actually regular points of class (1,α) for any 0 < α < 1.

In my talk I shall outline the theory of curvature for aribtrary closed sets and explain how it relates to the problem of regularity of varifolds.

Handwritten notes