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Let $V$ be an open bounded subset of an $n$-dimensional Riemannian manifold $(M,g)$ whose Gagliardo-Nirenberg constant satisfies\n  \\[\n  \\mathbb{G}_{\\alpha}^{\\pm}(V,g) \\geq \\mathbb{G}_{\\alpha}^{\\pm}(\\mathbb{R}^n,g_{\\mathbb{R}^n}),\n  \\]\n  where $(\\mathbb{R}^n,g_{\\mathbb{R}^n})$ denotes the $n$-dimensional Euclidean space with its standard metric. We show that for $\\alpha \\in (0,1) \\cup \\left(1,\\frac{n+6}{n+2}\\right)$ when $n \\leq 6$ or $\\alpha \\in (0,1) \\cup \\lef"},"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":"2507.05908","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-08T11:51:10Z","cross_cats_sorted":[],"title_canon_sha256":"a92742a8467fe8c65a7f436df89be05941d16ed70d3afd843617d8c572e33fff","abstract_canon_sha256":"4d2c4ff0f185d61583b59c46ee841ae0b114bdbb0cd2becda5ca95448e449c93"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:43:33.349988Z","signature_b64":"0qz0FSqK0NMMDVMyL10N6dsiuuN+zcSaR+VnwOZLgBclQcK+8QAmyuGk/4MEDDm/ulPb7b+BjvD1QDd7zj/LBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c405f1e5ee645ba4d464fc66b77f0754a22987dd3312d2c014e15f1504a2f33","last_reissued_at":"2026-07-05T11:43:33.349450Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:43:33.349450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rigidity of manifolds with respect to Gagliardo-Nirenberg inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Liang Cheng","submitted_at":"2025-07-08T11:51:10Z","abstract_excerpt":"In this paper, we investigate local rigidity properties related to Gagliardo-Nirenberg constants and unweighted Yamabe-type constants. Let $V$ be an open bounded subset of an $n$-dimensional Riemannian manifold $(M,g)$ whose Gagliardo-Nirenberg constant satisfies\n  \\[\n  \\mathbb{G}_{\\alpha}^{\\pm}(V,g) \\geq \\mathbb{G}_{\\alpha}^{\\pm}(\\mathbb{R}^n,g_{\\mathbb{R}^n}),\n  \\]\n  where $(\\mathbb{R}^n,g_{\\mathbb{R}^n})$ denotes the $n$-dimensional Euclidean space with its standard metric. We show that for $\\alpha \\in (0,1) \\cup \\left(1,\\frac{n+6}{n+2}\\right)$ when $n \\leq 6$ or $\\alpha \\in (0,1) \\cup \\lef"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05908","kind":"arxiv","version":2},"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/2507.05908/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":"2507.05908","created_at":"2026-07-05T11:43:33.349522+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.05908v2","created_at":"2026-07-05T11:43:33.349522+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05908","created_at":"2026-07-05T11:43:33.349522+00:00"},{"alias_kind":"pith_short_12","alias_value":"NRAF6HS64ZC3","created_at":"2026-07-05T11:43:33.349522+00:00"},{"alias_kind":"pith_short_16","alias_value":"NRAF6HS64ZC3UTKG","created_at":"2026-07-05T11:43:33.349522+00:00"},{"alias_kind":"pith_short_8","alias_value":"NRAF6HS6","created_at":"2026-07-05T11:43:33.349522+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/NRAF6HS64ZC3UTKGJ7DGW57QOV","json":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV.json","graph_json":"https://pith.science/api/pith-number/NRAF6HS64ZC3UTKGJ7DGW57QOV/graph.json","events_json":"https://pith.science/api/pith-number/NRAF6HS64ZC3UTKGJ7DGW57QOV/events.json","paper":"https://pith.science/paper/NRAF6HS6"},"agent_actions":{"view_html":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV","download_json":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV.json","view_paper":"https://pith.science/paper/NRAF6HS6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.05908&json=true","fetch_graph":"https://pith.science/api/pith-number/NRAF6HS64ZC3UTKGJ7DGW57QOV/graph.json","fetch_events":"https://pith.science/api/pith-number/NRAF6HS64ZC3UTKGJ7DGW57QOV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV/action/storage_attestation","attest_author":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV/action/author_attestation","sign_citation":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV/action/citation_signature","submit_replication":"https://pith.science/pith/NRAF6HS64ZC3UTKGJ7DGW57QOV/action/replication_record"}},"created_at":"2026-07-05T11:43:33.349522+00:00","updated_at":"2026-07-05T11:43:33.349522+00:00"}