Let $m\ge 2$, let $U\subset\mathbb R^m$ be open, and let $N$ be a compact smooth Riemannian manifold isometrically embedded in $\mathbb R^q$ for some $q\in\mathbb N$. Let $u\in W^{1,2}(U;N)$ be an energy-minimizing harmonic map, meaning that for every ball $B\Subset U$ and every competitor $v\in W^{1,2}(B;N)$ with $v-u\in W^{1,2}_0(B;\mathbb R^q)$, one has

\begin{align*} \int_B |\nabla u(x)|^2\,d\mathcal L^m(x)\le \int_B |\nabla v(x)|^2\,d\mathcal L^m(x). \end{align*}

Then $u$ is smooth outside a relatively closed singular set $\operatorname{sing}(u)$ with

\begin{align*} \mathcal H^{m-2}(\operatorname{sing}(u))=0. \end{align*}

Moreover, if $m\ge 3$, then the sharper estimate $\dim_{\mathcal H}\operatorname{sing}(u)\le m-3$ holds.