{"paper":{"title":"The $\\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\\frac{n}{2}$ and beyond","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Jonah A. J. Duncan, Luc Nguyen","submitted_at":"2025-07-22T09:38:29Z","abstract_excerpt":"Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\\geq 3$ with smooth non-empty boundary $\\partial M$. Let $\\Gamma\\subset\\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\\Gamma$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem\n  \\begin{align*}\n  \\begin{cases}\n  f(\\lambda(-g_u^{-1}A_{g_u})) = \\frac{1}{2}, \\quad \\lambda(-g_u^{-1}A_{g_u})\\in\\Gamma & \\text{on }M\\backslash \\partial M \\newline\n  u = 0 & \\text{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16394","kind":"arxiv","version":1},"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/2507.16394/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"}