Working with Mario Santilli on second order rectifiability of integral varifolds with bounded anisotropic first variation we tried to employ Allard’s results from his 1986 paper An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled. This turned out to be quite challenging. In particular, the conditions enumerated in 2.2(1)-(8) enforce the support of the varifold to have finite height inside an infinite cylinder. To apply the height bound from 3.5 (which yields uniform smallness of the height) one also needs bounded first variation and smallness of the \(\mathbf{L}^2\)-height. All these condition together are not, in general, satisfied at a generic point of an arbitrary integral varifold. The problem cannot be localised because cutting out a small ball either introduces unbounded first variation or violates a uniform height bound for the support in an infinite cylinder.

Trying to understand the proof of the height bound I was lured into the proof of 3.4(6), which is a kind of weak maximum principle. The proof ends with an enigmatic sentence “we arrive at a contradiction by choosing \(\tau\) less than but sufficiently close \(t_1\)”. This bugged me for several days so I decided to reach out for help. Finally, Guido De Philippis directed me to his paper written with Antonio De Rosa and Jonas Hirsch. Their proof of Theorem 3.4 is quite similar to Allard’s proof and very cleverly solves the issue. In the end I wrote my own version of Allard’s weak maximum principle.