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arXiv · 2605.17466

Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces

Anji Tang
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Abstract

The Schoen--Simon--Yau integral curvature estimate for stable minimal hypersurfaces is usually closed by Young's inequality. We give an alternative proof based on a single H\"older factorization. The choices \(f\mapsto f^{1+q}\) in the gradient estimate and \(\phi=u^{1+q}f^{1+q}\) in the stability inequality yield an explicit \(L^{4+2q}\)-estimate without negative powers of the cutoff. For strongly stable constant-mean-curvature hypersurfaces, with \(u=|\mathring A|\), the same factorization gives an estimate in which the cutoff scale and the mean-curvature scale remain separate. In the supercritical range \(2+q>n/2\), the Michael--Simon Sobolev inequality and Moser iteration turn this integral estimate into a quantitative pointwise bound. Its small-energy form is an \(\varepsilon\)-regularity statement.

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Anji Tang. 2026-05-17. Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces. https://arxiv.org/abs/2605.17466

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