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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.17466 |
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| _version_ | 1866913136958767104 |
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| author | Tang, Anji |
| author_facet | Tang, Anji |
| contents | 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 Hölder'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 Hölder 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 Hölder mechanism produces an integral curvature estimate featuring two competing terms, separated by the condition $|H|(1-θ)R\le 1$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Terminal Hölder Closure in Curvature Estimates and its Application Tang, Anji Differential Geometry 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 Hölder'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 Hölder 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 Hölder mechanism produces an integral curvature estimate featuring two competing terms, separated by the condition $|H|(1-θ)R\le 1$, 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. |
| title | Terminal Hölder Closure in Curvature Estimates and its Application |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.17466 |