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In dimensions n >= 3, we obtain a normalized O(r^{n-2}) growth estimate under the assumption that the fundamental group contains a free abelian subgroup of rank n-2. For locally conformally flat manifolds, we prove the corresponding normalized estimate outside the topological Euclidean case and derive polynomial or exponential upper bounds in the conformally Euclidean case.\n  In dimension three, under quadratic scal"},"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":"2606.01368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-31T17:52:34Z","cross_cats_sorted":[],"title_canon_sha256":"65f9ce6e02f76eaf2824e9a4c43cdf284c19c6deeea08a72bb83b2b5460c023d","abstract_canon_sha256":"5fd18dbbab121bde061defad5b31f0c0d935249be59728e5c102d6dee1b3c371"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:31.520842Z","signature_b64":"Wwpt5SQH6mFF1mi5eUtB3g9VndO+0C00e5nMLHTEbv7vePit8qChOsCUaSoUH2l7gGP14eGy5PXJ0yWvy3tVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7686f5b6f99ec33bb0ae9de77b839c7e84f359ed6943ad6cfb0dacc15a5713e1","last_reissued_at":"2026-06-02T02:04:31.520436Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:31.520436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jialong Deng","submitted_at":"2026-05-31T17:52:34Z","abstract_excerpt":"We prove Cohn-Vossen-type scalar-curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. 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