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We prove that if the lower bound $\\frac a2$ is achieved with multiplicity $k\\geq 1$, then $k\\leq n$, $M$ is isometric to $\\Sigma^{n-k}\\times \\mathbb{R}^k$ for some complete $(n-k)$-dimensional manifold $\\Sigma$ and by passing an isometr"},"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":"1305.4116","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-05-17T15:46:30Z","cross_cats_sorted":[],"title_canon_sha256":"2a8f5a0b52cf4654e205289b4c79e16862e45c88ebc574fe385b8d56dc9580f1","abstract_canon_sha256":"ced70b362f86110c4a32543f3733eae010b5aee37d101c7854b9899af1caa40e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:24.031549Z","signature_b64":"iUlQCR/NRuST4wFUaRc4SIRxPIt/X3o/qkhZ+blo51erdPB4yZHdUC9YLWVV/QavyrDALVNuCqlu3E1cIt9ECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2aa768c787728ce4eaa3863e30cdb10a2f7abe545dafcb71d5acdab1f3540f8c","last_reissued_at":"2026-05-18T03:10:24.030799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:24.030799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalues of the drifted Laplacian on complete metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Detang Zhou, Xu Cheng","submitted_at":"2013-05-17T15:46:30Z","abstract_excerpt":"I In this paper, first we study a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ with the ($\\infty$)-Bakry-\\'Emery Ricci curvature $\\textrm{Ric}_f\\ge \\frac a2g$ for some positive constant $a$. 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