The ${\sigma }_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\ge 3$ with smooth non-empty boundary $\mathrm{\partial }M$. Let $\mathrm{\Gamma }\subset {\mathbb{R}}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\mathrm{\Gamma }$ 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 \begin{align*} \begin{cases} f(λ(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad λ(-g_u^{-1}A_{g_u})\inΓ& $\text{on }$M\backslash \partial M \newline u = 0 & $\text{on }$\partial M \end{cases} \end{align*} admits a solution if ${\mu }_{\mathrm{\Gamma }}^{+}>1-\delta$, where ${\mu }_{\mathrm{\Gamma }}^{+}$ is defined by $(-{\mu }_{\mathrm{\Gamma }}^{+},1,\dots ,1)\in \mathrm{\partial }\mathrm{\Gamma }$ 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 $\lambda (-g^{-1}{A}_g)\in \mathrm{\Gamma }$ on $M$; these latter results require no assumption on ${\mu }_{\mathrm{\Gamma }}^{+}$ and are new when $(1,0,\dots ,0)\in \mathrm{\partial }\mathrm{\Gamma }$.