{"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. We prove a stability inequality and a Poincar\\'{e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01758","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":""},"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"}