$C^{\mathrm{\infty }}$ rectifiability of stationary varifolds

In this paper we prove that, for every integers $m\ge 2$ and $n\ge 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset {\mathbb{R}}^{m+n}$ is $C^{\mathrm{\infty }}$ rectifiable, namely it can be covered, up to an $H^m$-null set, with countably many $C^{\mathrm{\infty }}$ $m$-dimensional graphs.