{"paper":{"title":"A Liouville Theorem on the PDE $\\det(f_{i\\bar j})=1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"An-Min Li, Li Sheng","submitted_at":"2018-09-04T07:56:20Z","abstract_excerpt":"Let $f$ be a smooth plurisubharmonic function which solves $$ \\det(f_{i\\bar j})=1\\;\\;\\;\\;\\;\\;\\mbox{in }\\Omega\\subset \\mathbb C^n.$$ Suppose that the metric $\\omega_{f}=\\sqrt{-1}f_{i\\bar j}dz_{i}\\wedge d\\bar z_{j}$ is complete and $f$ satisfies the growth condition $$ C^{-1}(1+|z|^2)\\leq f\\leq C(1+ |z|^2),\\;\\;\\;\\; as\\;\\;\\; |z|\\to \\infty. $$ for some $C>0,$ then $f$ is quadratic."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00824","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"}