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Also we classify closed $f$-minimal hypersurfaces with $L_f$-index one immersed in $\\mathbb{S}^n(\\sqrt{2(n-1)})\\times \\mathbb{R}$ with the same $f$ as above."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.2379","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-05-10T16:13:56Z","cross_cats_sorted":[],"title_canon_sha256":"a6328712c63d7f2295868453cfc72c245ac4ef24dc33c3275fb7e7ae7eff0c8d","abstract_canon_sha256":"0f501e8423f6ad8790586e629225fc0c259e4623cdd49ccdf28c18e4d6661a33"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:00.154898Z","signature_b64":"huUucZjJJtsX3Yl4+EPnWpMDFxgQnFtCwTRd1l0xXCFEpj9uHd0TGzKBPToaD/I3xbn98LG7PbhzIj1NuHELCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce23bb63446020b3f5b4a8b0d1cafc7408fafe39693d9b5c57de186613e2c1c8","last_reissued_at":"2026-05-18T03:26:00.154298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:00.154298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simons' type equation for $f$-minimal hypersurfaces and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Detang Zhou, Tito Mejia, Xu Cheng","submitted_at":"2013-05-10T16:13:56Z","abstract_excerpt":"We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold $\\mathbb{S}^n(\\sqrt{2(n-1)})\\times \\mathbb{R}$ with $f=\\frac {t^2}{4}$. 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