Equivalence of intrinsic and extrinsic area bounds for minimal surfaces

TL;DR

This paper proves the equivalence of intrinsic and extrinsic area bounds for minimal surfaces and extends curvature estimates to the case of 6-dimensional minimal hypersurfaces, resolving a longstanding open problem.

Contribution

It establishes the equivalence of intrinsic and extrinsic area density bounds for all minimal immersions and extends Schoen--Simon--Yau curvature estimates to 6-dimensional cases.

Findings

Intrinsic and extrinsic area bounds are equivalent for minimal surfaces.

Extended curvature estimates to 6-dimensional stable minimal hypersurfaces.

Resolved an open case in minimal surface theory for n=6.

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.