{"paper":{"title":"About the Calabi problem: a finite dimensional approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CV","math.SG"],"primary_cat":"math.DG","authors_text":"H. -D. Cao, Julien Keller","submitted_at":"2011-02-05T20:41:56Z","abstract_excerpt":"Let us consider a projective manifold and $\\Omega$ a volume form. We define the gradient flow associated to the problem of $\\Omega$-balanced metrics in the quantum formalism, the \\Omega$-balacing flow. At the limit of the quantization, we prove that the $\\Omega$-balacing flow converges towards a natural flow in K\\\"ahler geometry, the $\\Omega$-K\\\"ahler flow. We study the existence of the $\\Omega$-K\\\"ahler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given K\\\"ahler class. We derive some natural geometric co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1097","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"}