{"paper":{"title":"The log-concavity of eigenfunction to complex Monge-Amp\\`ere operator in $\\mathbb{C}^2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qi Zhou, Wei Zhang","submitted_at":"2025-05-19T07:58:10Z","abstract_excerpt":"Following the authors' recent work \\cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Amp\\`ere operator. In this paper, we establish the $\\log$-concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Amp\\`ere operator on bounded, smooth, strictly convex domain in $\\mathbb{C}^2$. The key ingredients consist of the constant rank theorem and the deformation method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.12817","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.12817/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}