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This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy inequality \\[ {\\bf Ent}_{dv_g}(u) \\leq n \\log \\left(A_{opt} \\|D u\\|_{BV(M)} + B_{opt}\\right) \\] for all $u \\in BV(M)$ with $\\|u\\|_{L^1(M)} = 1$ and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent a optimal $L^1$-Sobolev inequality obtained by Druet [6]."},"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":"1404.3234","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2014-04-11T21:57:23Z","cross_cats_sorted":[],"title_canon_sha256":"0683d504b1d36e564d91388d2e0559a980af6b919d9aaaf4d77dd0788e1936fb","abstract_canon_sha256":"c3042825b5d210815ada54e72fb1a9e5d9af788a8b055dfe8f54d91cdb9092a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:41.604752Z","signature_b64":"JO9qIlmosn03x0ajV/LmE4OGDix+BlXRcWCnNV75A8CmqWdS1zC650w008ivOHJz06xNwYoN3fwryRVYhUfZAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bc7f6b6b392bcb8854d02cd8b09e6f1e4c1a56db83dc26d2db5f0c693a0237a","last_reissued_at":"2026-05-18T02:41:41.604063Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:41.604063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equivalence of optimal $L^1$-inequalities on Riemannian Manifolds","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jurandir Ceccon, Leandro Cioletti","submitted_at":"2014-04-11T21:57:23Z","abstract_excerpt":"Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \\geq 2$. 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