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In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth. In this paper, we prove the following theorem concerning harmonic functions on these manifolds.\n  Theorem: Let M be a complete noncompact manifold with nonnegative R"},"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":"math/9903172","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"1999-03-29T21:33:21Z","cross_cats_sorted":[],"title_canon_sha256":"fdbdc97c036e85cc44b3bbca725400bee435471a2009cfa24442c74cd22c0959","abstract_canon_sha256":"b8213c1882969eea32bd8cb2d952e496bd1cc45e23eb267e80f06de577550a9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:41:06.141206Z","signature_b64":"QizJGTaQ2bNDj0NUINQC8ZRp7a2c7cfeDqPwkFim26JsRwbjnwSb7NPAXlcYN66tk3wl3KLdnJT39Rd8PfSvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1fa250d39a8c42a273d66e0f32917f9e0dfc84b5463acf3a7f9661701a36911","last_reissued_at":"2026-07-04T14:41:06.140812Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:41:06.140812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth","license":"","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Christina Sormani","submitted_at":"1999-03-29T21:33:21Z","abstract_excerpt":"Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. 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