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More generally, for every \\(1<p<\\infty\\), the variational \\(p\\)-fundamental tone satisfies \\[\n  \\lambda_{1,p}(\\wti M)\n  \\geq\\left(\\frac{m-1}{p}\\right)^p, \\] and equality for some \\(p\\in(1,\\infty)\\) holds if and only if \\(\\wti M\\cong\\bH^m(-1)\\). In that case, equality holds for every \\(p\\in(1,\\infty)\\). The proof converts the two McKean defects of"},"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":"2608.00944","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-08-02T02:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"379bc3dbf585764369c0f51f5b3dae6a8d098c3d67a2878d3fc06b91b96e1ab6","abstract_canon_sha256":"fe4af1607f41f9beee9d8ce98168a01c92f9902f8ffb86a071b27902410d1664"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T01:55:53.763413Z","signature_b64":"a4aiAYpkDptReZuaCAfMeZm4RrBgSzQmeM8MX37b/C4WqiZIvOduyjuZ7eRz6R4pYmmKsjZSSuhnXR9LHMQ7Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dcfc47beb83e5c5ad8276b7583db112d1cb02e21457a7c638900afbaec77720","last_reissued_at":"2026-08-04T01:55:53.761927Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T01:55:53.761927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"McKean Rigidity for Cocompact Negatively Curved Manifolds and the \\(p\\)-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bo Zhu, Kuntao Jin","submitted_at":"2026-08-02T02:51:36Z","abstract_excerpt":"Let \\((M^m,g)\\) be a closed Riemannian manifold with \\(\\sec_g\\leq-1\\). We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature \\(-1\\). More generally, for every \\(1<p<\\infty\\), the variational \\(p\\)-fundamental tone satisfies \\[\n  \\lambda_{1,p}(\\wti M)\n  \\geq\\left(\\frac{m-1}{p}\\right)^p, \\] and equality for some \\(p\\in(1,\\infty)\\) holds if and only if \\(\\wti M\\cong\\bH^m(-1)\\). In that case, equality holds for every \\(p\\in(1,\\infty)\\). The proof converts the two McKean defects of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00944","kind":"arxiv","version":1},"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/2608.00944/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":"2608.00944","created_at":"2026-08-04T01:55:53.763174+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.00944v1","created_at":"2026-08-04T01:55:53.763174+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00944","created_at":"2026-08-04T01:55:53.763174+00:00"},{"alias_kind":"pith_short_12","alias_value":"PXH4I67LQPS4","created_at":"2026-08-04T01:55:53.763174+00:00"},{"alias_kind":"pith_short_16","alias_value":"PXH4I67LQPS4LLMC","created_at":"2026-08-04T01:55:53.763174+00:00"},{"alias_kind":"pith_short_8","alias_value":"PXH4I67L","created_at":"2026-08-04T01:55:53.763174+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/PXH4I67LQPS4LLMCO23VQPNRCL","json":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL.json","graph_json":"https://pith.science/api/pith-number/PXH4I67LQPS4LLMCO23VQPNRCL/graph.json","events_json":"https://pith.science/api/pith-number/PXH4I67LQPS4LLMCO23VQPNRCL/events.json","paper":"https://pith.science/paper/PXH4I67L"},"agent_actions":{"view_html":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL","download_json":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL.json","view_paper":"https://pith.science/paper/PXH4I67L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.00944&json=true","fetch_graph":"https://pith.science/api/pith-number/PXH4I67LQPS4LLMCO23VQPNRCL/graph.json","fetch_events":"https://pith.science/api/pith-number/PXH4I67LQPS4LLMCO23VQPNRCL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL/action/storage_attestation","attest_author":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL/action/author_attestation","sign_citation":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL/action/citation_signature","submit_replication":"https://pith.science/pith/PXH4I67LQPS4LLMCO23VQPNRCL/action/replication_record"}},"created_at":"2026-08-04T01:55:53.763174+00:00","updated_at":"2026-08-04T01:55:53.763174+00:00"}