Let $m\ge 2$, let $q\in\mathbb N$, and let $N\subset\mathbb R^q$ be a compact smooth embedded submanifold. There exist constants $\varepsilon_0>0$ and $C<\infty$, depending only on $m$ and $N$, with the following property. If $x_0\in\mathbb R^m$, $r>0$, and $u\in W^{1,2}(B(x_0,2r);N)$ is energy-minimizing with