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

Journal article

Let $(M^n , g_0)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ 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\[\left\{\begin{aligned}& f\big(\lambda(-g_u^{-1}A_{g_u})\big) = \tfrac12 , \quad && \lambda(-g_u^{-1}A_{g_u}) \in \Gamma \ \text{on } M\setminus\partial M, \\& u = 0, \quad && \text{on } \partial M\end{aligned}\right.\]admits a solution if $\mu^+_{\Gamma} > 1 - \delta$, where $\mu^+_{\Gamma}$ is defined by $(-\mu^+_{\Gamma}, 1, . . . , 1) \in \partial\Gamma$ and $\delta > 0$ is a constant depending on certain geometric data. In particular, we solve the $\sigma_k$-Loewner-Nirenberg problem for all $k \le \frac{n}{2}$, which extends recent work of the authors to include the important threshold case $k = \frac{n}{2}$. 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 $g \in [g_0]$ such that $\lambda(-g^{-1}A_g) \in \Gamma$ on $M$; these latter results require no assumption on $\mu^+_{\Gamma}$ and are new when $(1, 0, . . . , 0) \in \partial\Gamma$.

Authors: Duncan, JAJ; Nguyen, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Functional Analysis
• Volume: 290
• Issue: 6
• Article number: 111306
• Publication date: 2025-12-08
• Acceptance date: 2025-11-14
• DOI: 10.1016/j.jfa.2025.111306
• EISSN: 1096-0783
• ISSN: 0022-1236
• Language: English
• Keywords: elliptic PDE; fully nonlinear Yamabe-type problem; conformal geometry; complete metrics of negative curvature-typeC