Terminal H\"older Closure in Curvature Estimates and its Application

The Schoen--Simon--Yau (SSY) curvature estimate reduces the Bernstein problem for complete stable minimal graphs in $\mathbb{R}^{n+1}$ to an integral estimate whose final step traditionally relies on Young's inequality. This note shows that replacing Young's inequality by H\"older's inequality at this stage yields a structurally simpler argument, a strictly smaller constant, and a natural extension to the constant-mean-curvature (CMC) setting. Starting from the standard preparatory gradient estimate, we derive explicit constants $C_Y(n,q)$ and $C_H(n,q)$ for the Young and H\"older closure routes, and prove $\lim_{q\to0^+}C_H/C_Y=1/2$ with $C_H<C_Y$ for all sufficiently small $q$. For strongly stable CMC hypersurfaces, the same H\"older mechanism produces an integral curvature estimate featuring two competing terms, separated by the condition $|H|(1-\theta)R\le 1$, below this mean-curvature scale, the CMC estimate reduces to the minimal-surface form, quantitatively articulating that on scales smaller than its mean-curvature radius, a CMC hypersurface is locally indistinguishable from a stable minimal hypersurface.