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This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces.\n  Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\\rho = h/2$. Fixing a basepoint $o \\in X$, for $\\xi \\in \\partial X$, denote by $B_{\\xi}$ the Busemann function at $\\xi$ such that $B_{\\xi}(o) = 0$. then for $\\lambda \\in \\C$ the function $e^{(i\\lambda - \\rho)B_{\\xi}}$ is an eigenfunction of the Laplace-Beltrami operat"},"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":"1905.04112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-05-08T19:07:47Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"68b578d103d0a7cb4d7b97717a444d5076e3939f848464cf7624d23de6cdb15b","abstract_canon_sha256":"0f004c8ba3f48273cb29a4a56da9e2a9791cf4ca321e4d27c0d95e170c41cf1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:34.520224Z","signature_b64":"G/aiwp275wrySnv13NCwehwQZb2or2CW/WmR6DNblo3COEslHUD+NOsQl5ctZpbw+UBFE1RjJkjbM2iG0cIWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6800b9cc5f6b6ec5a8e488d2cd3745881be6df7476f0e23909a8bee5834cf580","last_reissued_at":"2026-05-17T23:46:34.519586Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:34.519586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Fourier transform on harmonic manifolds of purely exponential volume growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DG","authors_text":"Gerhard Knieper, Kingshook Biswas, Norbert Peyerimhoff","submitted_at":"2019-05-08T19:07:47Z","abstract_excerpt":"Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces.\n  Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\\rho = h/2$. Fixing a basepoint $o \\in X$, for $\\xi \\in \\partial X$, denote by $B_{\\xi}$ the Busemann function at $\\xi$ such that $B_{\\xi}(o) = 0$. then for $\\lambda \\in \\C$ the function $e^{(i\\lambda - \\rho)B_{\\xi}}$ is an eigenfunction of the Laplace-Beltrami operat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.04112","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":""},"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":"1905.04112","created_at":"2026-05-17T23:46:34.519697+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.04112v1","created_at":"2026-05-17T23:46:34.519697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.04112","created_at":"2026-05-17T23:46:34.519697+00:00"},{"alias_kind":"pith_short_12","alias_value":"NAALTTC7NNXM","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"NAALTTC7NNXMLKHE","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"NAALTTC7","created_at":"2026-05-18T12:33:24.271573+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/NAALTTC7NNXMLKHERDJM2N2FRA","json":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA.json","graph_json":"https://pith.science/api/pith-number/NAALTTC7NNXMLKHERDJM2N2FRA/graph.json","events_json":"https://pith.science/api/pith-number/NAALTTC7NNXMLKHERDJM2N2FRA/events.json","paper":"https://pith.science/paper/NAALTTC7"},"agent_actions":{"view_html":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA","download_json":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA.json","view_paper":"https://pith.science/paper/NAALTTC7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.04112&json=true","fetch_graph":"https://pith.science/api/pith-number/NAALTTC7NNXMLKHERDJM2N2FRA/graph.json","fetch_events":"https://pith.science/api/pith-number/NAALTTC7NNXMLKHERDJM2N2FRA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA/action/storage_attestation","attest_author":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA/action/author_attestation","sign_citation":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA/action/citation_signature","submit_replication":"https://pith.science/pith/NAALTTC7NNXMLKHERDJM2N2FRA/action/replication_record"}},"created_at":"2026-05-17T23:46:34.519697+00:00","updated_at":"2026-05-17T23:46:34.519697+00:00"}