Let $u_k:\Sigma\to N$ be a sequence of smooth harmonic maps from a closed Riemannian surface to a compact real-analytic target, with

\begin{align*} \sup_k E(u_k;\Sigma)<\infty. \end{align*}

Assume the following no-neck energy estimate for this target class, applied to the coordinate representatives of the maps in conformal charts. There is a constant $\varepsilon_{\mathrm{neck}}>0$ such that whenever $A_k=A(r_k,R_k)=\{z\in\mathbb{C}:r_k<|z|<R_k\}$ is a sequence of conformal annuli with $r_k/R_k\to 0$, and every dyadic subannulus $A(2^j r_k,2^{j+1}r_k)\subset A_k$ has energy at most $\varepsilon_{\mathrm{neck}}$ for all large $k$, then

\begin{align*} \lim_{k\to\infty}E(u_k;A_k)=0. \end{align*}

After passing to a subsequence, there are finitely many concentration points $p_1,\dots,p_J\in\Sigma$, a weak harmonic limit $u_\infty:\Sigma\to N$, and finitely many nonconstant harmonic spheres $\omega_{j\ell}:S^2\to N$ such that

\begin{align*} \lim_{k\to\infty}E(u_k;\Sigma) = E(u_\infty;\Sigma)+\sum_{j=1}^J\sum_\ell E(\omega_{j\ell};S^2). \end{align*}