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This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is $\\mathbb H(u)=(\\partial_j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincar\\'{e"},"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":"1506.01758","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-05T00:49:58Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"298bc2cd87cfd94d4b07b39bda75f5d9d122d1b6b85a39d6fd53de8364c78ff4","abstract_canon_sha256":"520d7c2e5ab33d2fe4736d81f9ee02101293efea410a2d20764ced5d90ff8bfb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:40.609646Z","signature_b64":"8Gav+c5LwwF7mxql3ZIOivQSil9gEdwpunMs63jjPIC2F8Hr8S5NILVoY6Zh0ChtzYaZOUQ5JAjxFTHDtkhoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a77e34d9040d2da608582b603969cc3d3962f852ba2fbe4f68b20c5d2f5cc05c","last_reissued_at":"2026-05-18T01:00:40.609178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:40.609178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stable solutions of symmetric systems on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2015-06-05T00:49:58Z","abstract_excerpt":"We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\\mathbb{M}$ without boundary, \\begin{equation*}\n  -\\Delta_g u_i = H_i(u_1,\\cdots,u_m) \\ \\ \\text{on} \\ \\ \\mathbb{M},\n  \\end{equation*} when $\\Delta_g$ stands for the Laplace-Beltrami operator, $u_i:\\mathbb{M}\\to \\mathbb R$ and $H_i\\in C^1(\\mathbb R^m) $ for $1\\le i\\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is $\\mathbb H(u)=(\\partial_j H_i(u))_{i,j=1}^m$, is symmetric. 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