{"paper":{"title":"Terminal H\\\"older Closure in Curvature Estimates and its Application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Replacing Young's inequality with Hölder's simplifies curvature estimates and extends to CMC hypersurfaces.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Anji Tang","submitted_at":"2026-05-17T14:07:53Z","abstract_excerpt":"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 rout"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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. 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 ≤ 1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The argument starts from the standard preparatory gradient estimate (cited as known) and assumes the hypersurface is strongly stable; if either the preparatory estimate fails or strong stability is weakened, the Hölder closure step does not apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Hölder closure of the SSY integral curvature estimate yields simpler arguments, strictly smaller constants than Young closure, and a quantitative reduction of CMC estimates to the minimal-surface case below a mean-curvature scale.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Replacing Young's inequality with Hölder's simplifies curvature estimates and extends to CMC hypersurfaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"13f76713f12054657b4510ea49b6b76cfcd09bacf9b5f6a3bc5870bbe724aaee"},"source":{"id":"2605.17466","kind":"arxiv","version":1},"verdict":{"id":"b53cdc09-5808-4577-a593-c9efccdbb369","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:37:30.108411Z","strongest_claim":"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. 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 ≤ 1.","one_line_summary":"Hölder closure of the SSY integral curvature estimate yields simpler arguments, strictly smaller constants than Young closure, and a quantitative reduction of CMC estimates to the minimal-surface case below a mean-curvature scale.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The argument starts from the standard preparatory gradient estimate (cited as known) and assumes the hypersurface is strongly stable; if either the preparatory estimate fails or strong stability is weakened, the Hölder closure step does not apply.","pith_extraction_headline":"Replacing Young's inequality with Hölder's simplifies curvature estimates and extends to CMC hypersurfaces."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17466/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.555839Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:51:37.536616Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.701266Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.657078Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"2ce821e6741175c524f9e0b2d7bfb73c4db7b46207ae3d6b9ad10961c437587a"},"references":{"count":11,"sample":[{"doi":"","year":1968,"title":"Simons,Minimal varieties in Riemannian manifolds, Ann","work_id":"608d8b4e-35dc-446b-84b4-a8c07b39eb38","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1975,"title":"R. 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