Almost-everywhere regularity of stationary integral varifolds

Theorem 1.1: every stationary integral $m$-varifold in an open subset of $\mathbb R^{m+n}$, $m,n\ge1$, has $\mathcal H^m(\mathrm{Sing}\,V)=0$, with no stability, minimizing, orientability, density-bound or codimension hypothesis. Corollary 1.2 gives the same on round spheres. The main new estimate is a signed excess bound in flat cylinders. The companion A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (5 Oct 2026) claims the stronger bound $\dim_{\mathcal H}\mathrm{Sing}\,V\le m-1$, which implies this one and is a separate entry; it uses this paper's signed-excess theorem as an input. Neither paper treats non-Euclidean ambient metrics, where De Lellis-Hirsch-Lihn-Spolaor construct exotic examples.

Theorem 1.1: every stationary integral $m$-varifold in an open subset of $\mathbb R^{m+n}$, $m,n\ge1$, has $\mathcal H^m(\mathrm{Sing}\,V)=0$, with no stability, minimizing, orientability, density-bound or codimension hypothesis. Corollary 1.2 gives the same on round spheres. The main new estimate is a signed excess bound in flat cylinders. The companion A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (5 Oct 2026) claims the stronger bound $\dim_{\mathcal H}\mathrm{Sing}\,V\le m-1$, which implies this one and is a separate entry; it uses this paper's signed-excess theorem as an input. Neither paper treats non-Euclidean ambient metrics, where De Lellis-Hirsch-Lihn-Spolaor construct exotic examples.