{"paper":{"title":"Mean field equations on a closed Riemannian surface with the action of an isometric group","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2018-11-27T14:49:45Z","abstract_excerpt":"Let $(\\Sigma,g)$ be a closed Riemannian surface, $\\textbf{G}=\\{\\sigma_1,\\cdots,\\sigma_N\\}$ be an isometric group acting on it. Denote a positive integer $\\ell=\\inf_{x\\in\\Sigma}I(x)$, where $I(x)$ is the number of all distinct points of the set $\\{\\sigma_1(x),\\cdots,\\sigma_N(x)\\}$. A sufficient condition for existence of solutions to the mean field equation $$\\Delta_g u=8\\pi\\ell\\left(\\frac{he^u}{\\int_\\Sigma he^udv_g}-\\frac{1}{{\\rm Vol}_g(\\Sigma)}\\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\\ell=1$ or equivalently $\\textbf{G}=\\{Id\\}$, where $Id$ is the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.11036","kind":"arxiv","version":1},"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"}