{"paper":{"title":"Stability properties and gap theorem for complete $f$-minimal hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Detang Zhou, Xu Cheng","submitted_at":"2013-07-18T23:44:14Z","abstract_excerpt":"In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\\mathbb{S}^n\\times \\mathbb{R}, \\bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\\mathbb{S}^n\\times\\{0\\}$ or $\\mathbb{S}^{n-1}\\times\\mathbb{R}$, where $\\mathbb{S}^{n-1}$ denotes the sphere in $\\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5099","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"}