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

Abstract

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ 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 \begin{align*} \begin{cases} f(\lambda(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma & \text{on }M\backslash \partial M \newline u = 0 & \text{on }\partial M \end{cases} \end{align*} admits a solution if $\mu_\Gamma^+ > 1-\delta$, where $\mu_\Gamma^+$ is defined by $(-\mu_\Gamma^+,1,\dots,1)\in\partial\Gamma$ and $\delta>0$ is a constant depending on certain geometric data. In particular, we solve the $\sigma_k$-Loewner-Nirenberg problem for all $k\leq \frac{n}{2}$, 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 $(1,0,\dots,0)\in\partial\Gamma$.