# Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

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.

• Look at the paper by Schoen, Simon, Yau. There's also a paper by Mike Anderson on minimal hypersurfaces. I'm pretty sure this estimate is in there. – Deane Yang Feb 22 '16 at 3:50

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).

• Dear Rbega, I found this note, but as it relies on previous estimates that don't seem "obviously true" outside of Euclidean space, I'm having a hard time following his proof. On another note, to make Choi-Schoen works, it seems one needs some favorable differential inequality derived from the Simon's Bochner formula for the second fundamental form, but there seems to be very little literature on this in general. In the end, I guess I'm going to have to get my hands dirty, but am I missing some silly? – Andy Sanders Feb 24 '16 at 19:19
• There are essentially two issues in these sorts of estimates: 1) Is there an $\epsilon$ sufficiently small so that if the total curvature is below this $\epsilon$ then the surface has pointwise bounded curvature and 2) Can one get an explicit bound on the pointwise curvature in terms of the total curvature. Point 1) is the more essential estimate and is a local statement and is most elegantly proved by blow up arguments as in the linked notes (note 8.12 holds in any Riemannian manifold). – Rbega Feb 24 '16 at 19:54
• The point being that to prove something like 1) it suffices to do most of your work in euclidean space as that is what the Riemannian manifold looks like on sufficiently small scales. – Rbega Feb 24 '16 at 19:58

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.

• (Side comment: I am no expert on this, so don't ask me about technical details.) – Willie Wong Feb 22 '16 at 14:22
• I vaguely recall that the SSY paper is pretty readable, and the proof not particularly hard by PDE standards. The proof is by Moser iteration, which uses only integration by parts, the Sobolev inequality, and the Holder inequality. As I've mentioned elsewhere, the key critical new idea (due, I believe, to Sacks-Uhlenbeck) is an improvement of Kato's inequality (which is really just Cauchy-Schwarz). Here, SSY's proof is enough but overly messy. It's easier to just work out your own proof by solving a finite-dimensional constrained minimization problem. – Deane Yang Feb 22 '16 at 14:33
• One comment. $\epsilon$-regularity of either the volume type (i.e., Allard's theorem) or total curvature type (i.e., a generalized Choi-Schoen theorem as in the question) both hold quite generally. The difficulties one encounters in developing the regularity theory in higher co-dimension lie elsewhere. The essential difficulty is to show that there are enough places where the hypotheses of the $\epsilon$-regularity results hold. – Rbega Feb 22 '16 at 15:53
• @Rbega points out an important point! Indeed that an $\epsilon$-regularity type result holds generally (see Theorem 4.2 in De Lellis's survey). The issue is more that the approach via $\epsilon$-regularity doesn't hold, due to that assumptions required to prove the necessary theorem does not hold sufficiently generally. – Willie Wong Feb 22 '16 at 16:33