Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces

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 Hölder factorization. The choices \(f\mapsto f^{1+q}\) in the gradient estimate and \(ϕ=u^{1+q}f^{1+q}\) 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.

20 pages