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The constant $\\frac{n-1}{n-2}$ cannot be improved, and if $\\mathrm{vol}(M)=\\mathrm{vol}(\\mathbb S^n)$ holds, then $M\\cong \\mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition.\n  The proofs i"},"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":"2405.08918","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-05-14T19:15:17Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"43e67a25888ba1258d7bf9d0fbccafbd0e62e35eae402beebf14e0abb886eabb","abstract_canon_sha256":"1192eb483e4817815ffe10954c5c4f0d4bad35601bcb49a013853b818fef8be8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:28:16.390923Z","signature_b64":"QJzMGpqPWzVKVz+57LwlyamQ/WPiOtC7wFfDGMO9wEErFcNTbpJ30nuuXs7W0v9Au3FHs1+zsdsnosTYc/OgBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bef702fb32da6f2b01c4d5926d1d8abba5465d257931fafcb7c586b6d4a0c7c","last_reissued_at":"2026-07-05T10:28:16.390377Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:28:16.390377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Gioacchino Antonelli, Kai Xu","submitted_at":"2024-05-14T19:15:17Z","abstract_excerpt":"We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\\geq3$ satisfies $$ \\lambda_1\\left(-\\frac{n-1}{n-2}\\Delta+\\mathrm{Ric}\\right)\\geq n-1, $$ then $\\operatorname{vol}(M)\\leq\\operatorname{vol}(\\mathbb S^{n})$, and $\\pi_1(M)$ is finite. The constant $\\frac{n-1}{n-2}$ cannot be improved, and if $\\mathrm{vol}(M)=\\mathrm{vol}(\\mathbb S^n)$ holds, then $M\\cong \\mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition.\n  The proofs i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.08918","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.08918/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2405.08918","created_at":"2026-07-05T10:28:16.390447+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.08918v3","created_at":"2026-07-05T10:28:16.390447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.08918","created_at":"2026-07-05T10:28:16.390447+00:00"},{"alias_kind":"pith_short_12","alias_value":"HPXXAL5TFWTP","created_at":"2026-07-05T10:28:16.390447+00:00"},{"alias_kind":"pith_short_16","alias_value":"HPXXAL5TFWTPFMA4","created_at":"2026-07-05T10:28:16.390447+00:00"},{"alias_kind":"pith_short_8","alias_value":"HPXXAL5T","created_at":"2026-07-05T10:28:16.390447+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":7,"internal_anchor_count":7,"sample":[{"citing_arxiv_id":"2605.14931","citing_title":"Spectral splitting theorem and ends of minimal hypersurfaces","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2412.12631","citing_title":"Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2602.11002","citing_title":"Simply connectedness of K\\\"ahler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)","ref_index":1,"is_internal_anchor":true},{"citing_arxiv_id":"2605.11384","citing_title":"Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2604.26529","citing_title":"Intermediate curvature and splitting theorem","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2604.06141","citing_title":"Finite index constant mean curvature hypersurfaces in low dimensions","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2604.14393","citing_title":"Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\\mathbb{R}^4$","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO","json":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO.json","graph_json":"https://pith.science/api/pith-number/HPXXAL5TFWTPFMA4JVMSNUOYVO/graph.json","events_json":"https://pith.science/api/pith-number/HPXXAL5TFWTPFMA4JVMSNUOYVO/events.json","paper":"https://pith.science/paper/HPXXAL5T"},"agent_actions":{"view_html":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO","download_json":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO.json","view_paper":"https://pith.science/paper/HPXXAL5T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.08918&json=true","fetch_graph":"https://pith.science/api/pith-number/HPXXAL5TFWTPFMA4JVMSNUOYVO/graph.json","fetch_events":"https://pith.science/api/pith-number/HPXXAL5TFWTPFMA4JVMSNUOYVO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO/action/storage_attestation","attest_author":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO/action/author_attestation","sign_citation":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO/action/citation_signature","submit_replication":"https://pith.science/pith/HPXXAL5TFWTPFMA4JVMSNUOYVO/action/replication_record"}},"created_at":"2026-07-05T10:28:16.390447+00:00","updated_at":"2026-07-05T10:28:16.390447+00:00"}