Rectifiability of every finite order for stationary integral varifolds

Abstract

We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral $d$ dimensional varifold in an open subset of $\mathbf{R}^n$ is $(\mathscr{H}^d, d)$ rectifiable of class $(k, \alpha)$ for every positive integer $k$ and every $0<\alpha<1$, and in particular of class $\mathscr{C}^{\infty}$. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at $\mathscr{H}^d$ almost every point, with the graph of a solution of the Euler--Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli's characterisation of higher order rectifiability together with Whitney's extension theorem.