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We show that the hyperbolic metric $g_0$ is stable in the following sense: if $g_i$ is a sequence of Riemaniann metrics on $M$ of same volume as $g_0$ and if $h(g_i)$ converges to $n-1$, then there are smooth subsets $Z_i\\subset M$ such that both $\\mathrm{Vol}(Z_i,g_i)$ and $\\m"},"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":"2302.07422","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-02-15T01:49:20Z","cross_cats_sorted":[],"title_canon_sha256":"432d64069a5db2ba26cae88e6fd61fec9403c1c408157fe131e599df80add9e5","abstract_canon_sha256":"6ab0c0b1c6a1def507ef42606364e89daa31282df7ace2b6fe38b37dd6de8046"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:41.950156Z","signature_b64":"7wKtJLN5d55We73J3H8zTPrQZPUckIm8EkGi8uf/8/ljHOFQ9dO2t/KCUnNrGFx2Qz78/pybftRNwyPRLXMTCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f7f61bbc9548a94d75e81600f3327b71df865e0ca55334ff33b414bdc63d612","last_reissued_at":"2026-07-05T12:00:41.949539Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:41.949539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Entropy and stability of hyperbolic manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Antoine Song","submitted_at":"2023-02-15T01:49:20Z","abstract_excerpt":"Let $(M,g_0)$ be a closed oriented hyperbolic manifold of dimension at least $3$. 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We show that the hyperbolic metric $g_0$ is stable in the following sense: if $g_i$ is a sequence of Riemaniann metrics on $M$ of same volume as $g_0$ and if $h(g_i)$ converges to $n-1$, then there are smooth subsets $Z_i\\subset M$ such that both $\\mathrm{Vol}(Z_i,g_i)$ and $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.07422","kind":"arxiv","version":5},"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/2302.07422/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":"2302.07422","created_at":"2026-07-05T12:00:41.949669+00:00"},{"alias_kind":"arxiv_version","alias_value":"2302.07422v5","created_at":"2026-07-05T12:00:41.949669+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.07422","created_at":"2026-07-05T12:00:41.949669+00:00"},{"alias_kind":"pith_short_12","alias_value":"H57WDO6JKSFJ","created_at":"2026-07-05T12:00:41.949669+00:00"},{"alias_kind":"pith_short_16","alias_value":"H57WDO6JKSFJJV26","created_at":"2026-07-05T12:00:41.949669+00:00"},{"alias_kind":"pith_short_8","alias_value":"H57WDO6J","created_at":"2026-07-05T12:00:41.949669+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.07864","citing_title":"On $3$-manifolds with small mass and $L^2$-curvature","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4","json":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4.json","graph_json":"https://pith.science/api/pith-number/H57WDO6JKSFJJV26QFQA6MZHW4/graph.json","events_json":"https://pith.science/api/pith-number/H57WDO6JKSFJJV26QFQA6MZHW4/events.json","paper":"https://pith.science/paper/H57WDO6J"},"agent_actions":{"view_html":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4","download_json":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4.json","view_paper":"https://pith.science/paper/H57WDO6J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2302.07422&json=true","fetch_graph":"https://pith.science/api/pith-number/H57WDO6JKSFJJV26QFQA6MZHW4/graph.json","fetch_events":"https://pith.science/api/pith-number/H57WDO6JKSFJJV26QFQA6MZHW4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4/action/storage_attestation","attest_author":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4/action/author_attestation","sign_citation":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4/action/citation_signature","submit_replication":"https://pith.science/pith/H57WDO6JKSFJJV26QFQA6MZHW4/action/replication_record"}},"created_at":"2026-07-05T12:00:41.949669+00:00","updated_at":"2026-07-05T12:00:41.949669+00:00"}