Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

Question

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.

Answer 1

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 of a course by Brian White (it's Theorem 8.12).

Answer 2

Deane's comment, which refers to this paper gives the basic answer for minimal hypersurfaces in arbitrary dimensions. (See also Wickramasekera's Annals paper for a recent vast general theory for regularity of minimal "hypersurfaces".)

For higher codimension: the short answer is that $\epsilon$-regularity doesn't work. (At least no one has figured out a sufficient version of it.) This is due to a very real obstruction. A very good summary of the status-quo for the regularity theory of minimal submanifolds is given by De Lellis in this survey article (alternate stable DOI link here); section 5 concerns precisely the question you asked.