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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 rout","authors_text":"Anji Tang","cross_cats":[],"headline":"Replacing Young's inequality with Hölder's simplifies curvature estimates and extends to CMC hypersurfaces.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-17T14:07:53Z","title":"Terminal H\\\"older Closure in Curvature Estimates and its Application"},"references":{"count":11,"internal_anchors":0,"resolved_work":11,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Simons,Minimal varieties in Riemannian manifolds, Ann","work_id":"608d8b4e-35dc-446b-84b4-a8c07b39eb38","year":1968},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"R. Schoen, L. Simon, and S.-T. 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