The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates

2025/07/22 by Jonah A. J. Duncan, Duncan, Jonah A. J., Luc Nguyen +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.16383

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

Abstract

In this paper we give a complete classification of positive viscosity solutions w to conformally invariant equations of the form \begincases f(λ(-A_w)) = (1)/(2), λ(-A_w)∈Γamp; in ℝ₊ⁿ \newline w = 0 amp; on ∂ℝ₊ⁿ, \endcases where A_w is the Schouten tensor of the metric g_w = w^-2|dx|², Γ⊂ℝⁿ is a symmetric convex cone and f is an associated defining function satisfying standard assumptions. Solutions to \eqrefab yield metrics g_w of negative curvature-type which are locally complete near ∂ℝ₊ⁿ. In particular, when (f,Γ) = (σ₁,Γ₁⁺), \eqrefab is the Loewner-Nirenberg problem in the upper half-space. More precisely, let μ_Γ⁺ denote the unique constant satisfying (-μ_Γ⁺, 1,…,1)∈∂Γ. We show that when μ_Γ⁺ >1 (e.g. when Γ= Γ_k⁺ for k<(n)/(2)), the hyperbolic solution w⁽⁰⁾(x) := x_n is the unique solution to \eqrefab. More surprisingly, we show that when μ_Γ⁺ ≤ 1 (e.g. when Γ= Γ_k⁺ for k≥ (n)/(2)), the solution set consists of a monotonically increasing one-parameter family \w^(a)(x_n)\_{a≥ 0}, of which the hyperbolic solution w⁽⁰⁾ is the minimal solution. In either case, solutions of \eqrefab are functions of x_n. Our proof involves a novel application of the method of moving spheres for which we must establish new estimates and regularity near ∂ℝ₊ⁿ, followed by a delicate ODE analysis. As an application, we give counterexamples to local boundary C⁰ estimates on solutions to the fully nonlinear Loewner-Nirenberg problem when μ_Γ⁺ ≤ 1.

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Cited by

• Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds
• The σ_k-Loewner-Nirenberg problem on Riemannian manifolds for k=(n)/(2) and beyond

Related

• The σ_k-Loewner-Nirenberg problem on Riemannian manifolds for k=(n)/(2) and beyond
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• Liouville theorems for conformally invariant fully nonlinear equations. I
• The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics