Let ${M}, \, {N}$ be two Riemannian manifolds. Let $u_n: {M} \to {N}$ be a sequence of harmonic maps. Suppose that $u_n$ converges uniformly to a continuous function $u$. Is $u$ a harmonic map?

Let ${M}, \, {N}$ be two Riemannian manifolds, and let $u_n: {M} \to {N}$ be a sequence of harmonic maps.

Question. Suppose that $u_n$ converges uniformly to a (necessarily continuous) function $u$. Is $u$ a harmonic map?

Let $\mathcal{M}, \mathcal{N}$ be two Riemannian manifolds. Let $u_n: \mathcal{M} \to \mathcal{N}$ be a sequence of harmonic maps. Suppose that $u_n$ converges uniformly to a continuous function $u$. Is $u$ a harmonic map?

Let ${M}, \, {N}$ be two Riemannian manifolds. Let $u_n: {M} \to {N}$ be a sequence of harmonic maps. Suppose that $u_n$ converges uniformly to a continuous function $u$. Is $u$ a harmonic map?