{"paper":{"title":"A new characterization for Clifford hypersurfaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Carlos Pe\\~nafiel, Qing Cui","submitted_at":"2024-03-04T03:29:58Z","abstract_excerpt":"For a closed minimal immersed hypersurface $M$ in $\\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\\ge 2$, define a constant $\\sigma_k=\\dfrac{\\int_M (|A|^2)^k}{|M|}$. We show that $\\sigma_k \\ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $\\sigma_2 \\ge 16$. When $n\\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\\ge 2$, there exists a positive constant $\\delta_k(n)<n$, if $|A|^2\\ge \\delta_k(n)$, we have $\\sigma_k\\ge n^k$. All the equality holds iff $M$ is isometric to a Cliff"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.01701","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.01701/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}