Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index

TL;DR

This paper establishes new measure bounds on the tubular neighborhood of the singular set of certain minimal hypersurfaces with finite index, improving previous bounds and providing deeper structural insights.

Contribution

It provides improved bounds on the measure of the singular set of minimal hypersurfaces with finite index, extending previous results and offering new structural understanding.

Findings

Bound on the upper Minkowski content of the singular set.

Improved measure bounds compared to previous work.

Enhanced structural insights into the singular set.

Abstract

We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral $n$-varifolds $V$ in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii) $\mathcal{H}^{n-1}$-null singular set. A direct consequence of such a bound is a bound on the upper Minkowski content of the singular set of such a varifold in terms of its total mass and its index. Such a result improves on known bounds, namely the corresponding bound on the $\mathcal{H}^{n-7}$-measure of the singular set established by A. Song ([10]), as well as the same bounds established by A. Naber and D. Valtorta ([6]) for codimension one area minimising currents. Our results also provide more structural information on the singular set for codimension one integral varifolds with finite index (on the regular part) and no classical singularities established by N. Wickramasekera ([11]).