{"paper":{"title":"New result on Chern conjecture for minimal hypersurfaces and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongwei Xu, Zhiyuan Xu","submitted_at":"2016-05-24T01:00:52Z","abstract_excerpt":"We verify that if $M$ is a compact minimal hypersurface in $\\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\\leq |A|^2-n\\leq\\frac{n}{22}$, then $|A|^2\\equiv n$ and $M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in $\\mathbb{R}^{n+1}$ whose equation is given by (\\ref{selfshr}), and if the squared length of the second fundamental form of $M$ satisfies $0\\leq|A|^2-1\\leq\\frac{1}{21}$, then $|A|^2\\equiv1$ and $M$ is a round sphere or a cylinder. Our results improve the rigidity theorems due to Q. Di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07250","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":""},"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"}