{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:F6ATTXLHZE74E6TAPMU3ECUFB2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"19ad1abc7d698440997ebe9a7326eec5cbfa9884aa0790ba383b77d5644b9b6d","cross_cats_sorted":[],"license":"","primary_cat":"math.DG","submitted_at":"2007-04-14T03:40:22Z","title_canon_sha256":"59e3a50869a5e3fb0da72ac177fcba9a5a8002873bd78e7481e3fd9bb5e8a8aa"},"schema_version":"1.0","source":{"id":"0704.1851","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.1851","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"arxiv_version","alias_value":"0704.1851v1","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.1851","created_at":"2026-05-18T04:27:02Z"},{"alias_kind":"pith_short_12","alias_value":"F6ATTXLHZE74","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"F6ATTXLHZE74E6TA","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"F6ATTXLH","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:a2b43302d9bd1b0d1816d22ec0d1280ac17c9e8917eda8ba79bd4061ba9c4361","target":"graph","created_at":"2026-05-18T04:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $M^{4n}$ be a complete quaternionic K\\\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\\lambda_1(M)$ of the Laplacian which is $\\lambda_1(M)\\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\\mathbb{R}\\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space $\\mathbb{QH}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When $\\lambda_1(M)\\ge \\frac{8(n+2)}3$, we also prove th","authors_text":"Detang Zhou, Peter Li, Shengli Kong","cross_cats":[],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2007-04-14T03:40:22Z","title":"Spectrum of the Laplacian on Quaternionic Kahler Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.1851","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d5a9195b2db3a5a7f187a49a2436c6ea2a0db5bd57084ccd1974704bafda3e6","target":"record","created_at":"2026-05-18T04:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"19ad1abc7d698440997ebe9a7326eec5cbfa9884aa0790ba383b77d5644b9b6d","cross_cats_sorted":[],"license":"","primary_cat":"math.DG","submitted_at":"2007-04-14T03:40:22Z","title_canon_sha256":"59e3a50869a5e3fb0da72ac177fcba9a5a8002873bd78e7481e3fd9bb5e8a8aa"},"schema_version":"1.0","source":{"id":"0704.1851","kind":"arxiv","version":1}},"canonical_sha256":"2f8139dd67c93fc27a607b29b20a850eb504da534266f69b8aad5022cd40c7a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f8139dd67c93fc27a607b29b20a850eb504da534266f69b8aad5022cd40c7a0","first_computed_at":"2026-05-18T04:27:02.868464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:02.868464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kFzwWzIRi3CUrKoEZiu0nkbTuIAGeytD/n0LDXnpof2anW1w4inlOqOT/kw/LUxxv000LzKJ0GvxkZzJC4X/Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:02.868976Z","signed_message":"canonical_sha256_bytes"},"source_id":"0704.1851","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d5a9195b2db3a5a7f187a49a2436c6ea2a0db5bd57084ccd1974704bafda3e6","sha256:a2b43302d9bd1b0d1816d22ec0d1280ac17c9e8917eda8ba79bd4061ba9c4361"],"state_sha256":"6144664fa6d7a895807acf22ffa5b38009d77ba8d2ba95c27ec3975e65e5e64a"}