For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.


Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (<http://link.springer.com/article/10.1007%2FBF01388577>)
for minimal surfaces in three-dimensional Riemannian manifolds extend to Riemannian manifold targets of arbitrary dimension? 


For non-experts, let me state the $\varepsilon$-regularity result of Choi and Schoen. Let $(M,h)$ be a three dimensional Riemannian manifold and suppose
\begin{align}
f: \Sigma \rightarrow M
\end{align}
is an immersed minimal surface; here the topology of $\Sigma$ is arbitrary. Let $(g,B)$ denote the induced metric and second fundamental form via the immersion $f.$ The following $\varepsilon$-regularity theorem posits that sufficient control on the $L^{2}$-norm of $B$ on balls of controlled size implies $L^{\infty}$-bounds on $B$ on balls of controlled size. 


**Theorem:** There exists constants $\varepsilon, \rho>0,$ (which depend only on the curvature of $M$ and it's covariant derivatives), such that if $r\_{0}<\rho, \ x\in M$ and
\begin{align}
f(\Sigma)\cap \partial B\_{r\_{0}}(x)=\partial(f(\Sigma)\cap B\_{r\_{0}}(x)),
\end{align}
the following property holds. If there exists $0<\delta\leq 1$ such that
\begin{align}
\int\_{B\_{r\_{0}}(x)\cap f(\Sigma)} \lVert B\rVert^{2} dV\_{g}< \delta\varepsilon,
\end{align}
then for all $0<\sigma\leq r\_{0}$ and all $y\in B\_{r\_{0}-\sigma}(x)\cap f(\Sigma)$,
\begin{align}
\sigma^{2}\lVert B\rVert^{2}(y)<\delta.
\end{align}


In the above theorem, $B\_{r}(x)$ is the Riemannian ball of radius $r>0$ in the manifold $M$ and $dV\_{g}$ is the Riemannian volume element of the metric $g$ on $\Sigma.$ Lastly, the symbol $\partial$ indicates the topological boundary of a set.


My question is if the manifold $M$ can be replaced by a Riemannian manifold of arbitrary dimension.


These types of regularity estimates date back to the work of Sacks-Uhlenbeck on minimal surfaces, and are an indispensable tool for the study of asymptotics to solutions of many systems of partial differential equations that have a geometric origin: examples include the Yang-Mills equations, the self-duality equations, the study of minimal surfaces, and more generally, harmonic maps. 


Unfortunately, for the most part I'm a casual observer to these developments and some basic google searches haven't pulled up any papers claiming this can be extended to arbitrary co-dimension. The obvious thing to do is to dive into the proof of Choi and Schoen, but before I undertake such a task, I was hoping someone here might save me the time. I appreciate any suggestions and/or references. Thank you.



Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds
The desired bound is correct (and in fact you get a fairly explicit value for $\epsilon$ of anything below $4\pi$) . A proof can be found in these beautiful [notes](http://web.stanford.edu/~cmad/Papers/MinSurfNotes.pdf)  of a course by Brian White (it's Theorem 8.12).


