{"paper":{"title":"Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger","submitted_at":"2011-03-09T16:15:13Z","abstract_excerpt":"Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\\geq-(n-1). The singular set S of Y has a stratification S^0\\subset S^1\\subset\\...\\subset S, where y\\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff dimension bound dimS^k\\leq k. Here, we define for all \\eta>0, 0<r\\leq 1, the {\\it k-th effective singular stratum} S^k_{\\eta,r} such that \\bigcup_\\eta\\bigcap_r \\,\\cS^k_{\\eta,r}= \\cS^k. Sharpening the bound dim S^k\\leq k, we prove that the r-tubular neighborhood satisfies: Vol(T_r("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1819","kind":"arxiv","version":4},"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"}