{
  "abstract": "We develop an \\(m\\)-Hessian approach to the construction of finite-Monge--Ampère weights on complete noncompact Kähler manifolds. Let \\((M^n,g)\\) be a complete noncompact Kähler manifold of complex dimension \\(n\\ge3\\) with positive holomorphic bisectional curvature. The main new ingredient is a quantitative capacity mechanism based on lower-order complex Hessian operators. More precisely, we obtain decay estimates for suitable relative \\(m\\)-Hessian capacities on dyadic annuli and show that these estimates imply the summability of the top-degree Monge--Ampère masses of a uniformly Lipschitz plurisubharmonic exhaustion. Consequently, we construct a proper function $$ u\\in PSH(M)\\cap C^{0,1}(M) $$ such that $$ \\int_M(dd^c u)^n<+\\infty. $$ The key point is the passage from lower-order \\(m\\)-Hessian capacity decay to finite Monge--Ampère mass, which is not a formal consequence of \\(m<n\\) Hessian mass estimates. We then explain how this finite-Monge--Ampère weight fits into the weighted holomorphic-function and analytic Bezout framework for uniformization. In particular, the construction provides a higher-dimensional pluripotential-theoretic mechanism that complements recent surface results and opens a route toward uniformization under positive curvature in complex dimensions \\(n\\ge3\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18171",
  "author_total": 1,
  "authors": [
    {
      "name": "Truong Dinh Dat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pgrgarpq2pc0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18171",
  "fulltext_url": null,
  "id": "6pgrgarpq2pc0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18171",
  "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.CV"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7213506211",
  "pdf_url": "https://arxiv.org/pdf/2609.18171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pgrgarpq2pc0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An $m$-Hessian approach to Yau uniformization conjecture",
  "title_text": "An m-Hessian approach to Yau uniformization conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/6pgrgarpq2pc0",
  "venue": "arXiv",
  "versions": []
}