Let $U \subset \mathbb{R}^n$ be open, let $1 \leq m \leq n$, and let $T$ be an $m$-dimensional integral current in $U$ that is locally area-minimizing in the following sense: for every compact set $K \subset U$ and every $m$-dimensional integral current $S$ in $U$ with $\operatorname{spt}(S - T) \subset K$ and $\partial S = \partial T$ in $U$, one has

\begin{align*} \mathsf{M}_K(T) \leq \mathsf{M}_K(S). \end{align*}

Suppose $x_0 \in \operatorname{spt} T \setminus \operatorname{spt}(\partial T)$ is an interior regular point of $T$. Then there exist $r > 0$, an integer $q \in \mathbb{N}$, and a smooth embedded oriented $m$-dimensional submanifold $M \subset B(x_0,r)$ such that $\overline{B}(x_0,r) \subset U$,

\begin{align*} T \llcorner B(x_0,r) = q \llbracket M \rrbracket, \end{align*}

and the mean curvature vector

\begin{align*} H_M: M \to \mathbb{R}^n \end{align*}

vanishes identically on $M$.