The σk-Loewner–Nirenberg problem on Riemannian manifolds for k <

Duncan, Jonah A. J.; Nguyen, Luc

doi:10.2140/apde.2025.18.2203

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The $\sigma_k$-Loewner–Nirenberg problem on Riemannian manifolds for $k\lt \frac{n}{2}$

Jonah A. J. Duncan and Luc Nguyen

Vol. 18 (2025), No. 9, 2203–2240

DOI: 10.2140/apde.2025.18.2203

Abstract

Let $(M^{n}, g_{0})$ be a smooth compact Riemannian manifold of dimension $n \geq 3$ with nonempty boundary $\partial M$ . Let $Γ \subset ℝ^{n}$ be a symmetric convex cone and $f$ a symmetric defining function for $Γ$ satisfying standard assumptions. Under an algebraic condition on $Γ$ , which is satisfied for example by the Gårding cones $Γ_{k}^{+}$ when $k < \frac{1}{2} n$ , we prove the existence of a locally Lipschitz viscosity solution $g_{u} = e^{2 u} g_{0}$ to the fully nonlinear Loewner–Nirenberg problem associated to $(f, Γ)$ ,

{\begin{matrix} f (λ (- g_{u}^{- 1} A_{g_{u}})) = 1, λ (- g_{u}^{- 1} A_{g_{u}}) \in Γ & on M ∖ \partial M, \\ u (x) \to + \infty & as {dist}_{g_{0}} (x, \partial M) \to 0, \end{matrix}

where $A_{g_{u}}$ is the Schouten tensor of $g_{u}$ . Previous results on Euclidean domains show that, in general, $u$ is not differentiable. The solution $u$ is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner–Nirenberg problems on approximating cones containing $(1, 0, \dots, 0)$ , for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric $g$ satisfying $λ (- g^{- 1} A_{g}) \in Γ$ on $M$ is a consequence of our results, rather than an assumption.

Keywords

Loewner–Nirenberg, fully nonlinear elliptic equations, negative curvature, complete conformal metrics, Yamabe

Mathematical Subject Classification

Primary: 35A01, 35A02, 35D40, 53C18, 53C21

Secondary: 35J60, 35J75

Milestones

Received: 31 October 2023

Revised: 6 September 2024

Accepted: 29 October 2024

Published: 5 September 2025

Authors

Jonah A. J. Duncan
Department of Mathematics University College London London United Kingdom

Luc Nguyen
Mathematical Institute University of Oxford Oxford United Kingdom

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Vol. 18, No. 9, 2025