{
  "abstract": "We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Ampère equation $\\det(u_{i\\bar{j}})=1$ with homogeneous Dirichlet data under the condition that, for some constant $κ\\geq 1$, $(κu_{i\\bar{j}}-u_{is}u^{s\\bar{t}}u_{\\overline{tj}})$ is non-negative definite. For $κ=1$, this gives the estimate for real convex solutions. The estimate depends only on the dimension, $κ$, and the $C^0$ and $C^1$ norms of the solution. As an application, every smooth entire convex solution of $\\det(u_{i\\bar{j}})=1$ on $\\mathbb{C}^n$ is a real quadratic polynomial. The same conclusion holds if real convexity is assumed only outside a compact subset of $\\mathbb{C}^n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18265",
  "author_total": 3,
  "authors": [
    {
      "name": "Hongyu Chen"
    },
    {
      "name": "Jingchen Hu"
    },
    {
      "name": "Li Sheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24xw7j9x2pa2t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18265",
  "fulltext_url": null,
  "id": "24xw7j9x2pa2t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18265",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18265",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24xw7j9x2pa2t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18265",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18265"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation",
  "title_text": "Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/24xw7j9x2pa2t",
  "venue": "arXiv",
  "versions": []
}