The σ_k-Loewner-Nirenberg problem on Riemannian manifolds for k=(n)/(2) and beyond

2025/07/22 by Jonah A. J. Duncan, Duncan, Jonah A. J., Luc Nguyen +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2507.16394

openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Mⁿ,g₀) be a smooth compact Riemannian manifold of dimension n≥ 3 with smooth non-empty boundary ∂ M. Let Γ⊂ℝⁿ be a symmetric convex cone and f a symmetric defining function for Γ satisfying standard assumptions. Denoting by A_gu the Schouten tensor of a conformal metric g_u = u^-2g₀, we show that the associated fully nonlinear Loewner-Nirenberg problem \begincases f(λ(-g_u^-1A_gu)) = (1)/(2), λ(-g_u^-1A_gu)∈Γamp; on M\backslash ∂ M \newline u = 0 amp; on ∂ M \endcases admits a solution if μ_Γ⁺ > 1-δ, where μ_Γ⁺ is defined by (-μ_Γ⁺,1,…,1)∈∂Γ and δ>0 is a constant depending on certain geometric data. In particular, we solve the σ_k-Loewner-Nirenberg problem for all k≤ (n)/(2), which extends recent work of the authors to include the important threshold case k=(n)/(2). In the process, we establish that the fully nonlinear Loewner-Nirenberg problem and corresponding Dirichlet boundary value problem with positive boundary data admit solutions if there exists a conformal metric g∈[g₀] such that λ(-g^-1A_g)∈Γ on M; these latter results require no assumption on μ_Γ⁺ and are new when (1,0,…,0)∈∂Γ.

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Cited by

• Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds
• Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature
• The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates

Related

• The σ_k-Loewner-Nirenberg problem on Riemannian manifolds for k<(n)/(2)
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• On Fully Nonlinear Loewner-Nirenberg Problem of Ricci curvature
• The Loewner-Nirenberg problem in singular domains