The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
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arXiv
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| Natura: | Preprint |
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2025
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| author | Duncan, Jonah A. J. Nguyen, Luc |
| author_facet | Duncan, Jonah A. J. Nguyen, Luc |
| contents | In this paper we give a complete classification of positive viscosity solutions $w$ to conformally invariant equations of the form
\begin{align}\label{ab}\tag{$*$}
\begin{cases}
f(λ(-A_w)) = \frac{1}{2}, \quad λ(-A_w)\inΓ& \text{in }\mathbb{R}_+^n \newline
w = 0 & \text{on }\partial\mathbb{R}_+^n,
\end{cases}
\end{align}
where $A_w$ is the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$, $Γ\subset\mathbb{R}^n$ is a symmetric convex cone and $f$ is an associated defining function satisfying standard assumptions. Solutions to \eqref{ab} yield metrics $g_w$ of negative curvature-type which are locally complete near $\partial\mathbb{R}_+^n$. In particular, when $(f,Γ) = (σ_1,Γ_1^+)$, \eqref{ab} is the Loewner-Nirenberg problem in the upper half-space.
More precisely, let $μ_Γ^+$ denote the unique constant satisfying $(-μ_Γ^+, 1,\dots,1)\in\partialΓ$. We show that when $μ_Γ^+ >1$ (e.g. when $Γ= Γ_k^+$ for $k<\frac{n}{2}$), the hyperbolic solution $w^{(0)}(x) := x_n$ is the unique solution to \eqref{ab}. More surprisingly, we show that when $μ_Γ^+ \leq 1$ (e.g. when $Γ= Γ_k^+$ for $k\geq \frac{n}{2}$), the solution set consists of a monotonically increasing one-parameter family $\{w^{(a)}(x_n)\}_{a\geq 0}$, of which the hyperbolic solution $w^{(0)}$ is the minimal solution. In either case, solutions of \eqref{ab} 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 $\partial\mathbb{R}_+^n$, followed by a delicate ODE analysis. As an application, we give counterexamples to local boundary $C^0$ estimates on solutions to the fully nonlinear Loewner-Nirenberg problem when $μ_Γ^+ \leq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates Duncan, Jonah A. J. Nguyen, Luc Analysis of PDEs Differential Geometry In this paper we give a complete classification of positive viscosity solutions $w$ to conformally invariant equations of the form \begin{align}\label{ab}\tag{$*$} \begin{cases} f(λ(-A_w)) = \frac{1}{2}, \quad λ(-A_w)\inΓ& \text{in }\mathbb{R}_+^n \newline w = 0 & \text{on }\partial\mathbb{R}_+^n, \end{cases} \end{align} where $A_w$ is the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$, $Γ\subset\mathbb{R}^n$ is a symmetric convex cone and $f$ is an associated defining function satisfying standard assumptions. Solutions to \eqref{ab} yield metrics $g_w$ of negative curvature-type which are locally complete near $\partial\mathbb{R}_+^n$. In particular, when $(f,Γ) = (σ_1,Γ_1^+)$, \eqref{ab} is the Loewner-Nirenberg problem in the upper half-space. More precisely, let $μ_Γ^+$ denote the unique constant satisfying $(-μ_Γ^+, 1,\dots,1)\in\partialΓ$. We show that when $μ_Γ^+ >1$ (e.g. when $Γ= Γ_k^+$ for $k<\frac{n}{2}$), the hyperbolic solution $w^{(0)}(x) := x_n$ is the unique solution to \eqref{ab}. More surprisingly, we show that when $μ_Γ^+ \leq 1$ (e.g. when $Γ= Γ_k^+$ for $k\geq \frac{n}{2}$), the solution set consists of a monotonically increasing one-parameter family $\{w^{(a)}(x_n)\}_{a\geq 0}$, of which the hyperbolic solution $w^{(0)}$ is the minimal solution. In either case, solutions of \eqref{ab} 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 $\partial\mathbb{R}_+^n$, followed by a delicate ODE analysis. As an application, we give counterexamples to local boundary $C^0$ estimates on solutions to the fully nonlinear Loewner-Nirenberg problem when $μ_Γ^+ \leq 1$. |
| title | The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2507.16383 |