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We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. we investigate the existence of positive distributional solutions u in C^0(M) to the critical equation \\Delta_g u + a(x) u = u^{2*(s)-1}/ d_g(x,x_0)^s in M where \\Delta_g := - div_g(\\nabla) is the Laplace-Beltrami operator, and d_g is the Riemannian distance on (M,g). Via a minimization method in the spirit of Aubin, we prove existence in dimension n > 3 when the potential a is sufficiently below the scalar curvatur"},"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":"1401.6133","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-01-23T19:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"45d8230029c13b060b97d66a16e8f9db14dc41b8784dccd19761ddb938017174","abstract_canon_sha256":"36c0ab0b2cf849861c98a00f47c1f61d86e3b3aff07ae4826bd51f015b550467"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:51.096599Z","signature_b64":"5fCeQn3DyvxSWChlXHyLW2m47bffo3UpG+fgln0gvfnL71Vyq/OLEVBQ1XpJmlMDPRJXVJ1PipDzUyJy3lVeBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b9220cdbb8745634dfc060f5c99ea1619d2973deb33ba160f3954f203d2008c","last_reissued_at":"2026-05-18T01:19:51.096040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:51.096040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hardy-Sobolev Equations on Compact Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hassan Jaber","submitted_at":"2014-01-23T19:11:58Z","abstract_excerpt":"Let (M,g) be a compact Riemannien Manifold of dimension n > 2, x_0 in M a fix and singular point and s in (0,2). 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