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

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) $$\mathcal {H}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -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 $$\mathcal {H}^{n-7}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msup> </mml:math> -measure of the singular set established by Song (Duke Math J 172(11):2073–2147, 2023), as well as the same bounds established by Naber and Valtorta (J Eur Math Soc 22(10):3305–3382, 2020) 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 Wickramasekera (Ann Math 179:843–1007, 2014).