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

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{$\ast$} \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 $g_w=w^{-2}\mathrm{\mid }dx{\mathrm{\mid }}^2$, $\mathrm{\Gamma }\subset {\mathbb{R}}^n$ 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 $\mathrm{\partial }{\mathbb{R}}_{+}^n$. In particular, when $(f,\mathrm{\Gamma })=({\sigma }_1,{\mathrm{\Gamma }}_1^{+})$, \eqref{ab} is the Loewner-Nirenberg problem in the upper half-space. More precisely, let ${\mu }_{\mathrm{\Gamma }}^{+}$ denote the unique constant satisfying $(-{\mu }_{\mathrm{\Gamma }}^{+},1,\dots ,1)\in \mathrm{\partial }\mathrm{\Gamma }$. We show that when ${\mu }_{\mathrm{\Gamma }}^{+}>1$ (e.g. when $\mathrm{\Gamma }={\mathrm{\Gamma }}_k^{+}$ 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 ${\mu }_{\mathrm{\Gamma }}^{+}\le 1$ (e.g. when $\mathrm{\Gamma }={\mathrm{\Gamma }}_k^{+}$ for $k\ge \frac{n}{2}$), the solution set consists of a monotonically increasing one-parameter family $\{w^{(a)}(x_n){\}}_{a\ge 0}$, 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 $\mathrm{\partial }{\mathbb{R}}_{+}^n$, 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 ${\mu }_{\mathrm{\Gamma }}^{+}\le 1$.