{"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 $a$. It is known that the spectrum of the drifted Laplacian $\\Delta_f$ for $M$ is discrete and the first nonzero eigenvalue of $\\Delta_f$ has lower bound $\\frac a2$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4116","kind":"arxiv","version":2},"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"}