Terminal H\"older Closure in Curvature Estimates for Stable Minimal and Strongly Stable CMC Hypersurfaces

We revisit the terminal step in the Schoen--Simon--Yau curvature estimate for stable minimal hypersurfaces. By applying the preliminary gradient estimate to the cutoff power \(f^{1+q}\), we obtain a direct Hölder closure estimate with an explicit constant. This argument avoids the negative powers of the cutoff which appear in the direct Young route.
We then extend the same closure mechanism to strongly stable constant mean curvature hypersurfaces in Euclidean space. For \(u=|\mathring A|\), the traceless second fundamental form, the mean curvature terms remain as lower-order perturbations. In particular, on balls satisfying \( |H|(1-\theta)R \le \left( \lambda \frac{\widetilde{\mathcal C}_1}{\widetilde{\mathcal C}_2} \right)^{\frac{1}{2+2q}}, \) the CMC estimate reduces to the minimal-type estimate up to the factor \(1+\lambda\).