{"paper":{"title":"A Liouville theorem for solutions of degenerate Monge-Amp\\`ere equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jingang Xiong, Tianling Jin","submitted_at":"2012-11-27T02:29:36Z","abstract_excerpt":"In this paper, we give a new proof of a celebrated theorem of J\\\"orgens which states that every classical convex solution of \\[ \\det\\nabla^2 u (x)=1\\quad {in} \\mathbb{R}^2 \\] has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Amp\\`ere equations. We prove that every convex generalized (or Alexandrov) solution of \\[ \\det \\nabla^2 u(x_1,x_2)=|x_1|^{\\alpha} \\quad {in} \\mathbb{R}^2, \\] where $\\alpha>-1$, has to be \\[ u(x_1,x_2)= \\frac{a}{(\\alpha+2)(\\alpha+1)}|x_1|^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6183","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"}