Equivalence of intrinsic and extrinsic area bounds for minimal surfaces

Abstract

We show that intrinsic and extrinsic area density bounds are equivalent, with matching asymptotic values, for complete, connected, smooth minimal immersions $i:{\Sigma }^d\to {\mathbb{R}}^N$ of any dimension and codimension. Combining our results with a recent breakthrough by Bellettini, we extend the Schoen--Simon--Yau curvature estimates for smoothly immersed, two-sided, stable minimal hypersurfaces $i:{\Sigma }^n\to {\mathbb{R}}^{n+1}$ with bounded intrinsic area density to the missing case $n=6$, which had remained open since.