Workshop on Geometric Analysis at Steinfeld Abbey on the occasion of Heiko’s von der Mosel 60th birthday

Abstract

The theory of varifolds developed by Almgren and Allard in the 1970s is a principal tool for studying solutions to geometric variational problems such as the Plateau problem. A 𝑘-dimensional varifold in an open set 𝑈 ⊆ 𝐑ⁿ is simply a Radon measure over the Cartesian product of 𝑈 with the Grassmannian 𝐆(𝑛,𝑘). It is suppose to model a 𝑘-dimensional surface together with information about its tangent planes. The notion of the mean curvature vector comes from the first variation of the area. However, one can also consider other functional, different from area, and this gives rise to a different notion of the “mean curvature”. In particular, given a non-Euclidean norm 𝜙 on 𝐑ⁿ one can use the 𝑘-dimensional Hausdorff measure ℋᵏ over 𝐑ⁿ which isn’t invariant under rotations or reflections (anisotropy).

Minimal surfaces are understood as varifolds 𝑉 with zero mean curvature. A priori these are only Radon measures so a regularity theory is being developed. In case 𝑉 is associated to a graph of some function 𝑢 : Rᵏ → 𝐑ⁿ⁻ᵏ one can apply PDE theory. However, the regularity theory for varifolds starts a few steps earlier: the goal is to show that around almost all points 𝑉 is associated to a graph. Actually, even mere rectifiability is an issue.

I shall present a summary of the theory of varifolds and brielfy describe the recent results concerning their rectifiabilty. After that, I shall describe my result obtained with Mario Santilli (not yet finished) on second order rectifiability of codimension one varifolds with bounded anisotropic mean curvature.

Handwritten notes