In their paper on the min-max construction of minimal surfaces, Colding and De Lellis make use of a compactness result for stable minimal surfaces. More precisely, given a stable minimal surface $\Sigma^2\subset U$ with $\partial \Sigma\subset \partial U$ and second fundamental form $A$ there is a curvature estimate $$ \vert A\vert^2(x) \le \frac{C}{\text{dist}(x,\partial U)^2} $$ valid for all $x\in \Sigma$. They state the curvature bound implies the following compactness property: whenever $\Sigma_i$ is a sequence of stable minimal surfaces in $U$ then there is a subsequence that converges to another stable minimal surface. I understand the curvature estimate, but I don't see how it implies the compactness result.

We can make the following local argument. Let $\Sigma_i$ be a sequence of stable minimal surfaces and let $p_i\in \Sigma_i$. Passing to a subsequence, we may assume that $p_i\to p$ and that the tangent planes $\pi_i$ to $\Sigma_i$ at $p_i$ converge to a plane $\pi$. The curvature bound implies that in a small neighborhood of $p$ the component of $\Sigma_i$ containing $p_i$ is graphical over $\pi$. We can then use Arzela-Ascoli to extract a convergent subsequence of these graphs. However, I don't see how to patch this together into a global argument. Passing to a subsequence, we can assume that $\Sigma_i\to \Sigma$ in the Hausdorff distance or as varifolds, but I don't see how to proceed from here. Could someone point me to a reference or explain how this is done?


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