{"paper":{"title":"Solutions of some Monge-Amp\\`ere equations with isolated and line singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Jingang Xiong, Tianling Jin","submitted_at":"2012-12-18T01:10:30Z","abstract_excerpt":"In this paper, we study existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Amp\\`ere equations with isolated and line singularities. We classify all solutions of $\\det \\nabla^2 u=1$ in $\\R^n$ with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having $k$ non-removable singular points for $k>1$, modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold of finite dimension. We also establish exi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4206","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"}