{"paper":{"title":"Sharp nonexistence results for curvature equations with four singular sources on rectangular tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Zhijie Chen","submitted_at":"2017-09-13T12:34:40Z","abstract_excerpt":"In this paper, we prove that there are no solutions for the curvature equation \\[ \\Delta u+e^{u}=8\\pi n\\delta_{0}\\text{ on }E_{\\tau}, \\quad n\\in\\mathbb{N}, \\] where $E_{\\tau}$ is a flat rectangular torus and $\\delta_{0}$ is the Dirac measure at the lattice points. This confirms a conjecture in \\cite{CLW2} and also improves a result of Eremenko and Gabrielov \\cite{EG}. The nonexistence is a delicate problem because the equation always has solutions if $8\\pi n$ in the RHS is replaced by $2\\pi \\rho$ with $0<\\rho\\notin 4\\mathbb{N}$. Geometrically, our result implies that a rectangular torus $E_{\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04287","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"}