A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2

TL;DR

This paper establishes a regularity and compactness theory for immersed stable minimal hypersurfaces with low multiplicity, showing they are locally representable as 2-valued C^{1, alpha} graphs under certain conditions.

Contribution

It introduces a new regularity result for stable minimal hypersurfaces near multiplicity 2 hyperplanes and provides applications like compactness and curvature estimates.

Findings

Hypersurfaces with singular sets of finite codimension are locally 2-valued C^{1, alpha} graphs.

A compactness theorem for immersed stable minimal hypersurfaces is established.

A pointwise curvature estimate in low dimensions is derived.

Abstract

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal hypersurfaces and a pointwise curvature estimate for the hypersurfaces in this class in low dimensions are also discussed.