{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:3FYBH3LATDLUNY7FWWXKPNPSZA","short_pith_number":"pith:3FYBH3LA","schema_version":"1.0","canonical_sha256":"d97013ed6098d746e3e5b5aea7b5f2c83fe6702a4ab8f2970b46d1041feadc93","source":{"kind":"arxiv","id":"1412.1326","version":2},"attestation_state":"computed","paper":{"title":"Topology and \\epsilon-regularity Theorems on Collapsed Manifolds with Ricci Curvature Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Ruobing Zhang","submitted_at":"2014-12-03T13:49:08Z","abstract_excerpt":"In this paper we discuss and prove $\\epsilon$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting.\n  A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\\ell},y^*)\\subset \\mathbb{R}^{n-\\ell}\\times C(Y^{\\ell-1})$ for $\\ell\\leq 3$, then in fact $|Rm|\\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy "},"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":"1412.1326","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-12-03T13:49:08Z","cross_cats_sorted":[],"title_canon_sha256":"a999998628e22ba29c4143651cb153ffff57cc4de0c7ce8b973ecacc67e8c5db","abstract_canon_sha256":"0547421c7360ac5ffcebd16a8e5fd8cb47eaa687b370ddc745e9bfe7acecd51a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:58.296493Z","signature_b64":"0BZtLk0S2s0KlYg8ZfciCjEyzg+K80tQL3SfLkl3W+N7+iuj8Cuf9Y6LCQDmeC4KhhuENrPnzirMA27B8kgJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d97013ed6098d746e3e5b5aea7b5f2c83fe6702a4ab8f2970b46d1041feadc93","last_reissued_at":"2026-05-18T01:01:58.295950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:58.295950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology and \\epsilon-regularity Theorems on Collapsed Manifolds with Ricci Curvature Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Ruobing Zhang","submitted_at":"2014-12-03T13:49:08Z","abstract_excerpt":"In this paper we discuss and prove $\\epsilon$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting.\n  A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\\ell},y^*)\\subset \\mathbb{R}^{n-\\ell}\\times C(Y^{\\ell-1})$ for $\\ell\\leq 3$, then in fact $|Rm|\\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1326","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1412.1326","created_at":"2026-05-18T01:01:58.296042+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.1326v2","created_at":"2026-05-18T01:01:58.296042+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1326","created_at":"2026-05-18T01:01:58.296042+00:00"},{"alias_kind":"pith_short_12","alias_value":"3FYBH3LATDLU","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"3FYBH3LATDLUNY7F","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"3FYBH3LA","created_at":"2026-05-18T12:28:11.866339+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA","json":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA.json","graph_json":"https://pith.science/api/pith-number/3FYBH3LATDLUNY7FWWXKPNPSZA/graph.json","events_json":"https://pith.science/api/pith-number/3FYBH3LATDLUNY7FWWXKPNPSZA/events.json","paper":"https://pith.science/paper/3FYBH3LA"},"agent_actions":{"view_html":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA","download_json":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA.json","view_paper":"https://pith.science/paper/3FYBH3LA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.1326&json=true","fetch_graph":"https://pith.science/api/pith-number/3FYBH3LATDLUNY7FWWXKPNPSZA/graph.json","fetch_events":"https://pith.science/api/pith-number/3FYBH3LATDLUNY7FWWXKPNPSZA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA/action/storage_attestation","attest_author":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA/action/author_attestation","sign_citation":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA/action/citation_signature","submit_replication":"https://pith.science/pith/3FYBH3LATDLUNY7FWWXKPNPSZA/action/replication_record"}},"created_at":"2026-05-18T01:01:58.296042+00:00","updated_at":"2026-05-18T01:01:58.296042+00:00"}